Question

In triangle KLM, if K is congruent to L, KL = 9x - 40, LM = 7x - 37, & KM = 3x + 23, find x & the measure of each angle.

Answer 1

Answer:

The value of x is 15. The measure of angle L and K is 45.7 degree. The  measure of angle M is 77.6 degree.

Step-by-step explanation:

It is given that

$\angle L\cong \angle K$

Since two angles are congruent, therefore we can say that the triangle KLM is isosceles triangle.

The sides KM and LM are congruent.

$3x+23=7x-37$

$60=4x$

$15=x$

The value of x is 15.

The length of side KL is

$KL=9x-40=9(15)-40=95$

The length of side KM and LM is

$KM=LM=7x-37=7(15)-37=68$

Therefore length of sides are 68, 68 and 95.

Use Law of cosine to find the measure of each angle.

$a^2=b^2+c^2-2bc\cos A$

Apply this formula according to the angle.

$L=K=\cos ^{-1}(\frac{(KM)^2+(KL)^2-(LM)^2}{2(KM)(KL)}) =\cos ^{-1}(\frac{68^2+95^2-68^2}{2(68)(95)})=45.69^{\circ}$

Therefore the measure of angle L and K is 45.7 degree.

The measure of M is calculated as,

$M=\cos ^{-1}(\frac{(LM)^2+(KM)^2-(KL)^2}{2(LM)(KM)}) =\cos ^{-1}(\frac{68^2+68^2-95^2}{2(68)(68)})=77.62^{\circ}$

Therefore the measure of angle M is 77.6 degree.

Answer 2

Answer:

x=15, ∠L=∠K=45.7° and ∠M is 77.6°.

Step-by-step explanation:

Given information: ∠L≅∠K, KL = 9x - 40, LM = 7x - 37, & KM = 3x + 23.

Two angles are congruent, so triangle KLM is an isosceles triangle. Corresponding adjacent sides of congruent angles are equal.

The sides KM and LM are congruent.

$3x+23=7x-37$

Isolate variable terms.

$3x-7x=-23-37$

$-4x=-60$

Divide both sides by -4.

$x=15$

The value of x is 15.

The length of sides of triangle KLM are

$KL=9x-40=9(15)-40=95$

$KM=LM=3(15)+23=45+23=68$

Therefore length of sides are 68, 68 and 95.

Law of cosine

$a^2=b^2+c^2-2bc\cos A$

Using the above formula we get

$K=L=\cos^{-1}(\frac{KM^2+KL^2-LM^2}{2(KM)(KL)})=\cos^{-1}(\frac{68^2+95^2-68^2}{2(68)(95)})=45.69^{\circ}$

$A=\cos^{-1}(\frac{LM^2+KM^2-KL^2}{2(LM)(KM)})=\cos^{-1}(\frac{68^2+68^2-95^2}{2(68)(68)})=77.62^{\circ}$

Therefore, ∠L=∠K=45.7° and ∠M is 77.6°.