4 THE ISOSCELES RIGHT TRIANGLEAN ISOSCELES RIGHT TRIANGLE is a standard mathematical object. The student should know the ratios of the sides. (An isosceles triangle has two equal sides. See Definition 8 in Some Theorems of Plane Geometry. The theorems cited below will be found there.) Theorem. In an isosceles right triangle the sides are in the ratio 1:1:
Proof. In an isosceles right triangle, the equal sides make the right angle. They are in the ratio 1 : 1. To find the ratio number of the hypotenuse h, we have, according to the Pythagorean theorem, h² = 1² + 1² = 2. Therefore, h = (Lesson 26 of Algebra.) And therefore the three sides are in the ratio 1 : 1 : Note that since the right triangle is isosceles, then the angles at the base are equal. (Theorem 3.) Therefore each of those acute angles is 45°. (For the definition of measuring angles by "degrees," see Topic 12.) Example 1. Evaluate sin 45° and tan 45°. Answer. For any problem involving 45°, the student should not consult the Table. Rather, sketch the triangle and place the ratio numbers. 
We see:
on rationalizing the denominator. (Lesson 26 of Algebra.)
Problem. Evaluate cos 45° and csc 45°.
Thus cos 45° is equal to sin 45°; they are complements.
Example 2. Solve the isosceles right triangle whose side is 6.5 cm. Answer. To solve a triangle means to know all three sides and all three angles. Since this is an isosceles right triangle, the only problem is to find the unknown hypotenuse. 
But in every isosceles right triangle, the sides are in the ratio 1 : 1 : Whenever we know the ratio numbers, we use this method of similar figures to solve the triangle, and not the trigonometric Table. (In Topic 6, we will solve right triangles the ratios of whose sides we do not know.)
Example 3. In an isosceles right triangle, the hypotenuse is Answer. The student should sketch the triangles and place the ratio numbers. 
How has the side corresponding to According to the rule for multiplying radicals, it has been multiplied by Next Topic: The 30°-60°-90° Triangle Please make a donation to keep TheMathPage online. Copyright © 2012 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |
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, as shown on the right. In the triangle on the left, the side corresponding to 1 has been multiplied by 6.5. Therefore every side will be multiplied by 6.5. The hypotenuse
will be 6.5
inches. How long are the sides?
. Therefore, all the sides will be multiplied by 