Question

PX and QY are altitudes of the acute triangle PQR, and Z is the midpoint of PQ. How do you prove that triangle XYZ is isosceles?

Answer 1

Using an assumption that RZ is also an altitude, ΔXYZ can be proven to be an isosceles triangle.

Response:

• Sides ZX and ZY of ΔXYZ are congruent by CPCTC, therefore, by definition ΔXYZ is an isosceles triangle

Which is the method used to prove that ΔXYZ is an isosceles triangle?

The given parameters are;

PX and QY are altitudes of triangle ΔPQR

The midpoint of PQ = Z

Required:

Whereby RZ is also an altitude, we have;

ΔQPY is similar to ΔRPZ by Angle-Angle similarity postulate.

ΔPQX ≅ ΔPQY by Angle-Angle-Side similarity postulate

PY ≅ QX by Corresponding Parts of Congruent Triangles are Congruent, CPCTC

• ΔZPY ≅ ΔZQX by Side-Angle-Side, SAS, congruency postulate

Therefore;

ZX ≅ ZY by CPCTC

• ΔXYZ is an isosceles triangle, by definition of isosceles triangles.

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