Chapter 10.7, Problem 7PSA

a)

To determine

If a rhombus is inscribed in a circle what must be true about the rhombus.

Answer to Problem 7PSA

Rhombus is a square

Explanation of Solution

Given Information: Rhombus ABCD is inscribed in a circle

Calculation:

  $\begin{array}{l}\text{For proving a rhombus is a square, we just need to prove that any one}\\ \text{ of its interior angles = 90° OR its diagonals are equal}\text{. Either of these}\text{.}\\ \text{PROOF :}\text{ diagonal DB is bisector of angle B \& angle D}\text{. (As ABCD}\\ \text{ is a rhombus, So triangle ABD is congruent to triangle CBD by SSS congruence criterion)}\\ \text{Now, 2a + 2b = 180° ( as, opposite angles of a cyclic quadrilateral are}\\ \text{always supplementary)}\\ \Rightarrow 2(a+b)=180°\\ \Rightarrow a+b=90°\\ \Rightarrow \text{In triangle ABD}\\ \Rightarrow \angle A\text{ = 180}-(a+b)\\ \Rightarrow \angle A=180-90=90°\\ \text{So, rhombus ABCD becomes a square}\text{.}\end{array}$

b)

To determine

If a trapezoid is inscribed in a circle what must be true about the trapezoid

Answer to Problem 7PSA

trapezoid is isosceles

Explanation of Solution

Given Information: Trapezoidal ABCD is inscribed in a circle

Calculation:

Let ABCD  be an isosceles trapezoid with the bases AB  and CD  and the                       

lateral sides AD  and BC.

We need to prove that there is a circle which passes through all the vertices of the trapezoid A, B, C  and D.

  $\begin{array}{l}\text{Let us draw the diagonals of the trapezoid AC and BD (Figure 1b) and}\\ \text{consider the triangles}\Delta ABC\text{ and }\Delta ABD.\end{array}$

These triangles have the common side AB  and the congruent sides BC  and 

AD  (the latest is because the trapezoid ABCD  is isosceles).

The angles BAD  and ABC  concluded between these congruent sides are congruent as the base angles of the isosceles trapezoid.

  $\begin{array}{l}\text{Hence, the triangles }\Delta ABC\text{ and }\Delta ABD\text{ABD are congruent in accordance with the SAS-test}\\ \text{for the triangles congruency}\text{.}\end{array}$

It implies that the angles ACB  and ADB  are congruent as the corresponding angles of congruent triangles. 

Thus the angles ACB  and ADB  are congruent and are leaning on the same segment AB. Hence, these angles are inscribed in a circle.

The converse statement is true that if the trapezoid is inscribed in a circle, then the trapezoid is isosceles. 

By combining the direct and the converse statements you can conclude that a trapezoid can be inscribed in a circle if and only if the trapezoid is isosceles.