Final answer:
To solve for x in the isosceles triangle XYZ where angles X and Z are congruent, we set XY equal to XZ and solve for x. The resulting side lengths are approximately 6.0833 units for both XY and XZ, and 10.6667 units for YZ.
Step-by-step explanation:
In ∆XYZ, with ∠X ≅ ∠Z, we have that XY = 13x – 21, YZ = 8x – 6, and XZ = x + 4. Since the triangle is isosceles with ∠X ≅ ∠Z, the legs opposite those angles are equal, thus XY = XZ. We set up the equation 13x – 21 = x + 4 and solve for x:
13x – x = 21 + 4
12x = 25
x = 25 / 12
x = 2.0833 (approx)
With the value of x, we find the lengths of the sides:
- XY = 13(2.0833) – 21 = 6.0833 (approx)
- YZ = 8(2.0833) – 6 = 10.6667 (approx)
- XZ = 2.0833 + 4 = 6.0833 (approx)
Finally, we find that ∆XYZ has side lengths approximately 6.0833 units, 10.6667 units, and 6.0833 units for XY, YZ and XZ respectively.