home Audience 4 The midline of a trapezoid is equal to the sum of its bases. How to find the midline of a trapezoid

The midline of a trapezoid is equal to the sum of its bases. How to find the midline of a trapezoid

The straight line segment connecting the midpoints of the lateral sides of the trapezoid is called the midline of the trapezoid. We will tell you below how to find the midline of a trapezoid and how it relates to other elements of this figure.

Centerline theorem

Let's draw a trapezoid in which AD is the larger base, BC is the smaller base, EF is the middle line. Let's extend the base AD beyond point D. Draw a line BF and continue it until it intersects with the continuation of the base AD at point O. Consider the triangles ∆BCF and ∆DFO. Angles ∟BCF = ∟DFO as vertical. CF = DF, ∟BCF = ∟FDО, because VS // JSC. Therefore, triangles ∆BCF = ∆DFO. Hence the sides BF = FO.

Now consider ∆ABO and ∆EBF. ∟ABO is common to both triangles. BE/AB = ½ by condition, BF/BO = ½, since ∆BCF = ∆DFO. Therefore, triangles ABO and EFB are similar. Hence the ratio of the parties EF/AO = ½, as well as the ratio of the other parties.

We find EF = ½ AO. The drawing shows that AO = AD + DO. DO = BC as sides of equal triangles, which means AO = AD + BC. Hence EF = ½ AO = ½ (AD + BC). Those. the length of the midline of a trapezoid is equal to half the sum of the bases.

Is the midline of a trapezoid always equal to half the sum of the bases?

Let us assume that there is a special case where EF ≠ ½ (AD + BC). Then BC ≠ DO, therefore, ∆BCF ≠ ∆DCF. But this is impossible, since they have two equal angles and sides between them. Therefore, the theorem is true under all conditions.

Midline problem

Suppose, in our trapezoid ABCD AD // BC, ∟A = 90°, ∟C = 135°, AB = 2 cm, diagonal AC is perpendicular to the side. Find the midline of the trapezoid EF.

If ∟A = 90°, then ∟B = 90°, which means ∆ABC is rectangular.

∟BCA = ∟BCD - ∟ACD. ∟ACD = 90° by convention, therefore, ∟BCA = ∟BCD - ∟ACD = 135° - 90° = 45°.

If in a right triangle ∆ABC one angle is equal to 45°, then the legs in it are equal: AB = BC = 2 cm.

Hypotenuse AC = √(AB² + BC²) = √8 cm.

Let's consider ∆ACD. ∟ACD = 90° according to the condition. ∟CAD = ∟BCA = 45° as the angles formed by the transversal of the parallel bases of the trapezoid. Therefore, legs AC = CD = √8.

Hypotenuse AD = √(AC² + CD²) = √(8 + 8) = √16 = 4 cm.

Midline of trapezoid EF = ½(AD + BC) = ½(2 + 4) = 3 cm.

The concept of the midline of the trapezoid

First, let's remember what kind of figure is called a trapezoid.

Definition 1

A trapezoid is a quadrilateral in which two sides are parallel and the other two are not parallel.

In this case, the parallel sides are called the bases of the trapezoid, and the non-parallel sides are called the lateral sides of the trapezoid.

Definition 2

The midline of a trapezoid is a segment connecting the midpoints of the lateral sides of the trapezoid.

Trapezoid midline theorem

Now we introduce the theorem about the midline of a trapezoid and prove it using the vector method.

Theorem 1

The midline of the trapezoid is parallel to the bases and equal to their half-sum.

Proof.

Let us be given a trapezoid $ABCD$ with bases $AD\ and\ BC$. And let $MN$ be the middle line of this trapezoid (Fig. 1).

Figure 1. Midline of trapezoid

Let us prove that $MN||AD\ and\ MN=\frac(AD+BC)(2)$.

Consider the vector $\overrightarrow(MN)$. We next use the polygon rule to add vectors. On the one hand, we get that

On the other side

Let's add the last two equalities and get

Since $M$ and $N$ are the midpoints of the lateral sides of the trapezoid, we will have

We get:

Hence

From the same equality (since $\overrightarrow(BC)$ and $\overrightarrow(AD)$ are codirectional and, therefore, collinear) we obtain that $MN||AD$.

The theorem has been proven.

Examples of problems on the concept of the midline of a trapezoid

Example 1

The lateral sides of the trapezoid are $15\ cm$ and $17\ cm$ respectively. The perimeter of the trapezoid is $52\cm$. Find the length of the midline of the trapezoid.

Solution.

Let us denote the midline of the trapezoid by $n$.

The sum of the sides is equal to

Therefore, since the perimeter is $52\ cm$, the sum of the bases is equal to

So, by Theorem 1, we get

Answer:$10\cm$.

Example 2

The ends of the circle's diameter are $9$ cm and $5$ cm away from its tangent, respectively. Find the diameter of this circle.

Solution.

Let us be given a circle with center at point $O$ and diameter $AB$. Let's draw a tangent $l$ and construct the distances $AD=9\ cm$ and $BC=5\ cm$. Let's draw the radius $OH$ (Fig. 2).

Figure 2.

Since $AD$ and $BC$ are the distances to the tangent, then $AD\bot l$ and $BC\bot l$ and since $OH$ is the radius, then $OH\bot l$, therefore, $OH |\left|AD\right||BC$. From all this we get that $ABCD$ is a trapezoid, and $OH$ is its midline. By Theorem 1, we get

The concept of the midline of the trapezoid

First, let's remember what kind of figure is called a trapezoid.

Definition 1

A trapezoid is a quadrilateral in which two sides are parallel and the other two are not parallel.

In this case, the parallel sides are called the bases of the trapezoid, and the non-parallel sides are called the lateral sides of the trapezoid.

Definition 2

The midline of a trapezoid is a segment connecting the midpoints of the lateral sides of the trapezoid.

Trapezoid midline theorem

Now we introduce the theorem about the midline of a trapezoid and prove it using the vector method.

Theorem 1

The midline of the trapezoid is parallel to the bases and equal to their half-sum.

Proof.

Let us be given a trapezoid $ABCD$ with bases $AD\ and\ BC$. And let $MN$ be the middle line of this trapezoid (Fig. 1).

Figure 1. Midline of trapezoid

Let us prove that $MN||AD\ and\ MN=\frac(AD+BC)(2)$.

Consider the vector $\overrightarrow(MN)$. We next use the polygon rule to add vectors. On the one hand, we get that

On the other side

Let's add the last two equalities and get

Since $M$ and $N$ are the midpoints of the lateral sides of the trapezoid, we will have

We get:

Hence

From the same equality (since $\overrightarrow(BC)$ and $\overrightarrow(AD)$ are codirectional and, therefore, collinear) we obtain that $MN||AD$.

The theorem has been proven.

Examples of problems on the concept of the midline of a trapezoid

Example 1

The lateral sides of the trapezoid are $15\ cm$ and $17\ cm$ respectively. The perimeter of the trapezoid is $52\cm$. Find the length of the midline of the trapezoid.

Solution.

Let us denote the midline of the trapezoid by $n$.

The sum of the sides is equal to

Therefore, since the perimeter is $52\ cm$, the sum of the bases is equal to

So, by Theorem 1, we get

Answer:$10\cm$.

Example 2

The ends of the circle's diameter are $9$ cm and $5$ cm away from its tangent, respectively. Find the diameter of this circle.

Solution.

Let us be given a circle with center at point $O$ and diameter $AB$. Let's draw a tangent $l$ and construct the distances $AD=9\ cm$ and $BC=5\ cm$. Let's draw the radius $OH$ (Fig. 2).

Figure 2.

First sign

If two sides and an angle two sides and a corner

Second sign

Third sign

The two circles are concentric

Proof.

Let A 1 A 2... A n be a given convex polygon, and n >

Parallelogram

Parallelogram

Properties of a parallelogram

opposite sides are equal;
opposite angles are equal;

d 1 2 +d 2 2 =2(a 2 +b 2).

Trapezoid

Trapeze

reasons and non-parallel sides - sides. middle line.

The trapezoid is called isosceles(or isosceles

rectangular.

Properties of a trapezoid

Signs of a trapezoid

Rectangle

Rectangle

Rectangle Properties

all properties of a parallelogram;
diagonals are equal.

Rectangle signs

1. One of its angles is straight.

2. Its diagonals are equal.

Rhombus

Diamond

Properties of a rhombus

all properties of a parallelogram;
diagonals are perpendicular;

Signs of a diamond

Square

Square

Properties of a square

all corners of a square are right;

Signs of a square

Signs of a parallelogram

Middle line

Theorem.

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.

Median

Median of a triangle is a segment connecting the vertex of a triangle with the middle of the opposite side of this triangle.

Formulas for the area of a rhombus

S = a 2 sin α

Trapezoid area formulas

S = 1(a + b) h

Formulas for the area of a circle

Formula for the arc of a circle and its length

L=2Pr L=Pr /180

First sign

If two sides and an angle between them one triangle are respectively equal two sides and a corner there is another triangle between them, then such triangles are congruent.

Second sign

If side and two adjacent angles one triangle are respectively equal side and two adjacent angles another triangle, then such triangles are congruent.

Third sign

If three sides of one triangle are respectively equal to three sides of another triangle, then such triangles are congruent.

A circle is a figure that consists of all points of the plane that are equidistant from a given point.

This point (O) is called the center of the circle.

The distance (r) from a point on a circle to its center is called the radius of the circle.

A radius is also called any segment connecting a point on a circle to its center.

A chord is a segment connecting two points on a circle.

The chord passing through the center of the circle is called the diameter (d=2r).

Tangent - a straight line (a) passing through a point (A) of a circle perpendicular to the radius drawn to this point is called.

In this case, this point (A) of the circle is called the point of tangency.

The part of the plane bounded by a circle is called a circle.

A circular sector is a part of a circle lying inside the corresponding central angle.

A circular segment is a common part of a circle and a half-plane, the boundary of which contains the chord of this circle.

The two circles are concentric(that is, having a common center) if and only if

Segments of tangents to a circle drawn from one point are equal and make equal angles with a straight line passing through this point and the center of the circle.

A tangent to a circle is perpendicular to the radius drawn to the point of tangency.

Two lines in a plane are called parallel if they do not intersect.

Theorem 1: if, when two lines intersect with a transversal, the lying angles are equal, then the lines are parallel.

Theorem 2: if, when two straight lines intersect with a transversal, the sum of the internal one-sided angles is equal to 180°, then the straight lines are parallel.

Theorem 3: if, when two lines intersect with a transversal, the corresponding angles are equal, then the lines are parallel:

Two lines parallel to a third are parallel.

Through a point not lying on a given line, one and only one line can be drawn parallel to the given one.

If two parallel lines are intersected by a third line, then the intersecting interior angles are equal.

If two parallel lines are intersected by a third line, then the corresponding angles are equal.

If two parallel lines are intersected by a third line, then the sum of the interior one-sided angles is 180°.

Theorem on the sum of angles of a convex polygon

For a convex n-gon, the sum of the angles is 180°(n-2).

Proof.

To prove the theorem on the sum of the angles of a convex polygon, we use the already proven theorem that the sum of the angles of a triangle is equal to 180 degrees.

Let A 1 A 2... A n be a given convex polygon, and n > 3. Let us draw all the diagonals of the polygon from the vertex A 1. They divide it into n – 2 triangles: Δ A 1 A 2 A 3, Δ A 1 A 3 A 4, ... , Δ A 1 A n – 1 A n . The sum of the angles of a polygon is the sum of the angles of all these triangles. The sum of the angles of each triangle is 180°, and the number of triangles is (n – 2). Therefore, the sum of the angles of a convex n-gon A 1 A 2... A n is equal to 180° (n – 2).

The sum of the angles in any triangle is 180°.

Proof. Consider triangle ABC and draw a line parallel to AC through vertex B (see figure). We have ÐKBM = ÐBAC, since these angles are corresponding angles formed by the intersection of parallel CA and BM by the secant AB. Angles ACB and CBM are also equal, since the angle vertical to ÐCBM is the corresponding angle for Ð ACB (here the secant is CB). Thus, Ð CAB + Ð ACB + Ð ABC = Ð MBK + ÐMBC + Ð ABC = 180°.

A leg of a right triangle opposite an angle of 30° is equal to half the hypotenuse

Theorem. The exterior angle of any triangle is greater than every interior angle of a triangle that is not adjacent to it.

Parallelogram

Parallelogram is a quadrilateral whose opposite sides are parallel in pairs.

Properties of a parallelogram

opposite sides are equal;
opposite angles are equal;
the diagonals are divided in half by the intersection point;
the sum of angles adjacent to one side is 180°;
the sum of the squares of the diagonals is equal to the sum of the squares of all sides:

d 1 2 +d 2 2 =2(a 2 +b 2).

Trapezoid

Trapeze is a quadrilateral in which two opposite sides are parallel and the other two are non-parallel.

The parallel sides of a trapezoid are called its reasons and non-parallel sides - sides. The segment connecting the midpoints of the sides is called middle line.

The trapezoid is called isosceles(or isosceles), if its sides are equal.

A trapezoid, one of whose angles is right, is called rectangular.

Properties of a trapezoid

its midline is parallel to the bases and equal to their half-sum;
if the trapezoid is isosceles, then its diagonals are equal and the angles at the base are equal;
if the trapezoid is isosceles, then a circle can be described around it;
If the sum of the bases is equal to the sum of the sides, then a circle can be inscribed in it.

Signs of a trapezoid

A quadrilateral is a trapezoid if its parallel sides are not equal

Rectangle

Rectangle is called a parallelogram in which all angles are right.

Rectangle Properties

all properties of a parallelogram;
diagonals are equal.

Rectangle signs

A parallelogram is a rectangle if:

1. One of its angles is straight.

2. Its diagonals are equal.

Rhombus

Diamond is called a parallelogram in which all sides are equal.

Properties of a rhombus

all properties of a parallelogram;
diagonals are perpendicular;
the diagonals are the bisectors of its angles.

Signs of a diamond

1. A parallelogram is a rhombus if:

2. Its two adjacent sides are equal.

3. Its diagonals are perpendicular.

4. One of the diagonals is the bisector of its angle.

Square

Square is called a rectangle whose sides are all equal.

Properties of a square

all corners of a square are right;
The diagonals of the square are equal, mutually perpendicular, the point of intersection bisects and bisects the corners of the square.

Signs of a square

A rectangle is a square if it has any characteristics of a rhombus.

Signs of a parallelogram

A quadrilateral is a parallelogram if:

1. Its two opposite sides are equal and parallel.

2. Opposite sides are equal in pairs.

3. Opposite angles are equal in pairs.

4. The diagonals are divided in half by the point of intersection.

The midline of a triangle is the segment connecting the midpoints of its two sides.

The middle line of a triangle connecting the midpoints of two given sides is parallel to the third side and equal to half of it.

Middle line A trapezoid is a segment connecting the midpoints of the sides of the trapezoid.

The midline of the trapezoid is parallel to the bases of the trapezoid and equal to their half-sum.

The locus of points that have a certain property is the set of all points that have this property.

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