Question

Suppose that ABCD is a trapezoid, with AB parallel to CD, and diagonals AC and BD intersecting at P. Explain why (a) triangles ABC and ABD have the same area; (b) triangles BCP and DAP have the same area; (c) triangles ABP and CDP are similar; (d) triangles BCP and DAP need not be similar.

Answer 1

Triangles ABC and ABD have the same area due to a shared base and altitude, as do triangles BCP and DAP. While triangles ABP and CDP are similar due to shared angles and parallel sides, triangles BCP and DAP do not necessarily share these properties and therefore need not be similar.

Step-by-step explanation:

When studying the properties of trapezoid ABCD with parallel sides AB and CD, and intersecting diagonals AC and BD at point P, we can draw several conclusions:

1. The triangles ABC and ABD share base AB and have the same altitude from point D and C respectively to side AB. Hence, they have the same area.
2. Triangles BCP and DAP also have the same area since they share base PD and have the same altitude from A and C respectively to PD.
3. Considering the similar triangles ABP and CDP, we know they are similar since they share the angle at P, and parallel sides AB and CD imply the corresponding angles at A and C are equal.
4. Finally, triangles BCP and DAP need not be similar since their corresponding sides and angles may not be in proportion.

Answer 2

A trapezoid ABCD is show below,

AB || CD

AC and BD are diagonals intersecting at P.

(a)

In Δ ABC and ABD,

Area of triangle ABC = Area of triangle ABD

Because,

Area of triangle ABC = 1/2 × Base × Height

Base = AB

Height = h, which is same for both the triangle that's why,

Area of triangle ABC = Area of triangle ABD

(b)

As we know that,

Area of triangle ABC = Area of triangle ABD

On subtracting the area of triangle ABP from both the triangles, we get

Area of triangle ABC - area of triangle ABP = Area of triangle ABD - area of triangle ABP

⇒ area of triangle BCP = area of triangle DAP

Hence, triangles BCP and DAP have the same area.

(c)

In triangle ABP and CDP,

∠ABP = ∠CDP (opposite interior angles are equal between two parallel lines)

∠APB = ∠CPD (opposite angles are equal)

∠BAP = ∠DCP (opposite interior angles are equal between two parallel lines)

Therefore,

By AAA similarity

Triangles ABP and CDP are similar.

(d)

In triangle BCP and DAP,

Only ∠BCP = ∠APD (opposite angles are equal).

Therefore,

Triangles BCP and DAP need not be similar.