Trigonometry

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5

THE ISOSCELES RIGHT TRIANGLE

AN ISOSCELES RIGHT TRIANGLE is one of two special triangles. (The other is the 30°-60°-90° triangle.) 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:square root of 2.

An isosceles right triangle

Proof.  In an isosceles right triangle, the equal sides make the right angle. They have the ratio of equality, 1 : 1.

To find the ratio number of the hypotenuse h, we have, according to the Pythagorean theorem,

h2 = 12 + 12 = 2.

Therefore,

h = square root of 2.

(Lesson 26 of Algebra.)  Therefore the three sides are in the ratio

1 : 1 : square root of 2.

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.

An isosceles right triangle

We see:

sin 45°   =   1 
square root of 2
 = ½square root of 2,

on rationalizing the denominator. (Lesson 26 of Algebra.)

tan 45°  =  1
1
 = 1.

Problem.   Evaluate cos 45° and csc 45°.

An isosceles triangle cos 45° =   1 
square root of 2
 = ½square root of 2.

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

csc 45° =  square root of 2
 1
 = square root of 2.

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.

An isosceles right triangle

But in every isosceles right triangle, the sides are in the ratio 1 : 1 : square root of 2, 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 6.5square root of 2.  (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 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 square root of 10 inches.  How long are the sides?

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

An isosceles right triangle

How has the side corresponding to square root of 2 been multiplied?

According to the rule for multiplying radicals, it has been multiplied by square root of 5.  Therefore, all the sides will be multiplied by square root of 5.  And 1timessquare root of 5 = square root of 5.


Next Topic:  Solving Right Triangles


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