# Chapter 23: Chapter 23

Problem 469

Let the congruent sides of an isosceles triangle have lengths \(\mathrm{r}\), and let the included angle have measure \(\theta\). Prove that the area enclosed by the triangle is given by \(\mathrm{A}=(1 / 2) \mathrm{r}^{2} \sin \theta\).

Problem 471

A parallelogram whose base is represented by \(\mathrm{x}+4\) and whose altitude is represented by \(\mathrm{x}-1\) is equivalent to a square whose side is 6 . Find the base and altitude of the parallelogram.

Problem 472

The lengths of two consecutive sides of a parallelogram are 10 inches and 15 inches, and these sides include an angle of \(63^{\circ}\). (As shown in the figure), a) Find, to the nearest tenth of an inch, the length of the altitude drawn to the longer side of the parallelogram, b) Find, to the nearest square inch, the area of the parallelogram.

Problem 475

Find the altitude AF of a rhombus if the lengths of the diagonals are 10 and 24 .

Problem 476

Find the area of a rhombus, each of whose sides is 10 in., and one of whose diagonals is 16 in.

Problem 477

The area of a rhombus is 90 , and one diagonal is 10 . Find the length of the other diagonal.

Problem 479

A trapezoid has two bases, of 6 and 10 Inches, and a height of 1 foot. What is its area?

Problem 481

Find the area of trapezoid \(\mathrm{ABCD}\), as shown in the diagram, if the length of base \(\underline{\mathrm{AB}}=9\) in., the length of base \(\underline{\mathrm{DC}}=5\) in., and the length of altitude $\underline{D E}=6$ in.

Problem 482

The wall at one end of an attic takes on the shape of a trapezoid because of a slanted ceiling. The wall is 8 feet high at one end, 10 feet wide, and only 3 feet high at the other end. What is the wall's area?