Question

Prove that “The medians to the legs of an isosceles triangle are congruent.”

Given: Isosceles triangle ABC with AB ≅ AC BT and CS are the medians
Prove: BT ≅ CS

Answer 1

Answer:

To prove that the medians to the legs of an isosceles triangle are congruent, we need to show that BT ≅ CS.

Proof:
We will start by drawing a diagram of the given isosceles triangle ABC with medians BT and CS.

[Insert a diagram of the isosceles triangle ABC with medians BT and CS]

Since AB ≅ AC, we know that angles B and C are congruent. Thus, we have:

∠BAC = ∠ABC = ∠ACB

Next, we will draw a line through point A parallel to BC and label the intersection point with BT as D.

[Insert a diagram of the isosceles triangle ABC with line AD parallel to BC]

Since AD is parallel to BC, we know that ∠BAD = ∠ACB (alternate interior angles) and ∠ADB = ∠ABC (corresponding angles). We also know that BD = DC (since BT is a median), so we have:

△ABD ≅ △ACD (by angle-angle-side)

Therefore, we have:

AD ≅ AD (reflexive property)

BD ≅ DC (given)

AB ≅ AC (given)

Thus, by the triangle congruence, we have BT ≅ CS (corresponding parts of congruent triangles are congruent).

Therefore, we have proved that the medians to the legs of an isosceles triangle are congruent.

Answer 2

Explanation:

We know that (a−b)2=a2+b2−2ab. . Thus, we have AC=AB. Hence, we have proved that if two medians of a triangle are equal, then the triangle is isosceles.