Question

A, B, C, and D are noncoplanar. $\mathrm{\Delta }\mathbf{ABC}\mathbf{,}\mathbf{}\mathrm{\Delta }\mathbf{ACD}$and $\mathrm{\Delta }\mathbf{ABD}$are equilateral. X and Y are midpoints of $\overline{\mathrm{AC}}\mathrm{and}\overline{\mathrm{AD}}$. Z is a point on $\overline{\mathrm{AB}}$. What kind of triangle is $\mathrm{\Delta }\mathbf{XYZ}$? Explain.

Short answer

$\mathrm{\Delta }XYZ$ is anisosceles triangle.

Step-by-step solution

Step 1. Description of step.

Consider$\mathrm{\Delta }XAZ$ and $\mathrm{\Delta }YAZ$. From the given information, $\overline{AZ}\cong \overline{AZ}$,$\angle XAZ\cong \angle YAZ$ and$\overline{AY}\cong \overline{AX}$ then by SAS postulate, $\mathrm{\Delta }XAZ\cong \mathrm{\Delta }YAZ$.

Step 2. Description of step.

By side angle side postulate$\mathrm{\Delta }ABC\cong \mathrm{\Delta }ACD$ as they are equilateral triangles. Hence, the corresponding parts$\overline{XY}$ and$\overline{YZ}$ of congruent triangles$\mathrm{\Delta }XAZ$and$\mathrm{\Delta }YAZ$are congruent.

Therefore,$\mathrm{\Delta }XYZ$ is an isosceles triangle.

Step 3. Write a paragraph proof.

$\mathrm{\Delta }XAZ\cong \mathrm{\Delta }YAZ$by SAS postulate.$\mathrm{\Delta }ABC\cong \mathrm{\Delta }ACD$by SAS postulate. The corresponding parts$\overline{XY}$and$\overline{YZ}$ of congruent triangles$\mathrm{\Delta }XAZ$ and$\mathrm{\Delta }YAZ$are congruent.

Therefore,$\mathrm{\Delta }XYZ$is an isosceles triangle.