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6

THE ISOSCELES RIGHT TRIANGLE

AN 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 also 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 = .

And therefore the three sides are in the ratio 1 : 1 : .

Note that since the triangle is isosceles, then the angles at the base are equal. (Theorem 3.)  Therefore each of those acute angles is 45°.

Example 1.   Evaluate sin 45° and tan 45°.

Answer.  For any problem involving 45°, the student should not consult the Table but, rather, should sketch the triangle and place the ratio numbers.

We see:

sin 45°

=

1

= ½,

on rationalizing the denominator.

Problem.   Evaluate cos 45° and csc 45°.

cos 45° =

1

= ½.

cos 45° is equal to sin 45°; they are complements.

csc 45° =

1

= .

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 : , 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.  (The theorem of the same multiple.)

Whenever we know the ratio numbers, we use this method of similar figures to solve the triangle, and not the trigonometric Table.

(In Topic 8, we will solve right triangles the ratios of whose sides we do not know.)

Example 3.   In an isosceles right triangle, the hypotenuse is inches.  How long are the sides?

Answer.  The student should sketch the triangles and place the ratio numbers.

How has the side corresponding to been multiplied?

According to the rule for multiplying radicals, it has been multiplied by .  Therefore, all the sides will be multiplied by .  And 1 = .

Next Topic:  The 30°-60°-90° Triangle

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