Question

1. in △klm, if ∠k≅∠l, kl = 9x - 40, lm = 7x - 37, and km = 3x + 23, find x and the measure of each side.
2. in △xyz if ∠x≅∠z, xy = 13x - 21, yz = 8x - 6, and xz = x + 4, find x and the measure of each side.
3. if △jkl is an equilateral triangle, jk = 13x + 5, kl = 17x - 19, and jl = 8x + 35, find x and the measure of each side.

Explanation:

Step1: Use the property of isosceles triangle

In $\triangle KLM$, since $\angle K\cong\angle L$, then $KL = LM$ (sides opposite to congruent angles in an isosceles triangle are equal). So we set up the equation $9x - 40=7x - 37$.
$9x-7x=- 37 + 40$
$2x=3$
$x=\frac{3}{2}$

Step2: Find the length of each side

• For $KL$: Substitute $x = \frac{3}{2}$ into $KL=9x - 40$.

$KL=9\times\frac{3}{2}-40=\frac{27}{2}-40=\frac{27 - 80}{2}=-\frac{53}{2}$ (This is incorrect. Let's correct the first - step equation setup. Since $\angle K\cong\angle L$, we should have $KM = LM$).
Set up the correct equation $3x + 23=7x - 37$.
$7x-3x=23 + 37$
$4x=60$
$x = 15$

• For $KL$: $KL=9x - 40=9\times15-40=135 - 40 = 95$
• For $LM$: $LM=7x - 37=7\times15-37=105 - 37 = 68$
• For $KM$: $KM=3x + 23=3\times15+23=45 + 23 = 68$

For the second triangle $\triangle XYZ$:

Step1: Use the property of isosceles triangle

Since $\angle X\cong\angle Z$, then $XY = YZ$ (sides opposite to congruent angles in an isosceles triangle are equal). Set up the equation $13x-21=8x - 6$.
$13x-8x=-6 + 21$
$5x=15$
$x = 3$

Step2: Find the length of each side

• For $XY$: $XY=13x - 21=13\times3-21=39 - 21 = 18$
• For $YZ$: $YZ=8x - 6=8\times3-6=24 - 6 = 18$
• For $XZ$: $XZ=x + 4=3+4 = 7$

Answer:

For $\triangle KLM$:
$x = 15$
$KL = 95$
$LM = 68$
$KM = 68$
For $\triangle XYZ$:
$x = 3$
$XY = 18$
$YZ = 18$
$XZ = 7$