# Chapter 23: Chapter 23

Problem 452

Find the area of an equilateral triangle whose perimeter is 24 .

Problem 453

Find the length of a side of an equilateral triangle whose area is $4 \sqrt{3}$.

Problem 455

Show that the median of any triangle separates the triangle into two regions of equal area.

Problem 460

The semi perimeter s of a triangle of sides \(a, b\), and \(c\) is defined as half the perimeter, or, \((a+b+c) / 2\). Find an expression for the area of \(\triangle \mathrm{ABC}\) and show that it is equivalent to \(\sqrt{[s(s-a)(s-b)(s-c)]}\).

Problem 461

Show that the ratio of the areas of two similar triangles equals the square of their ratio of similitude.

Problem 462

If two triangles have congruent bases, then the ratio of their areas equals the ratio of the lengths of their altitudes.

Problem 463

Triangle \(\mathrm{ABC}\) is similar to triangle $\mathrm{A}^{\prime} \mathrm{B}^{\prime} \mathrm{C}^{\prime} .\( If \)\mathrm{BC}=4$ and \(\mathrm{B}^{\prime} \mathrm{C}^{\prime}=12\), find the ratio of the areas of the triangles.

Problem 464

If the ratio of a pair of corresponding altitudes of two similar triangles is \(6 / 7\), and the area of the larger triangle is 98, find the area of the smaller triangle.

Problem 465

The areas of two similar triangles are in the ratio of \(4: 1\). The length of a side of the smaller triangle is \(5 .\) Find the length of the corresponding side in the larger triangle.

Problem 468

Let the two congruent sides of an isosceles triangle have lengths \(\mathrm{a}\), and let the included angle have measure \(\theta\). Choosing the third side as a base, let the corresponding altitude be of length \(\mathrm{r}\). Prove that the area enclosed by the triangle is given by the formula \(\mathrm{A}=\mathrm{r}^{2} \tan \theta / 2\)