how to solve problems in pythagorean theorem

Trigonometry : The Pythagorean Theorem

Pythagorean Theorem, states that the square of the longest side of a right triangle, is equal to the sum of the squares of the other two sides. Based on the formula given above, it is a such. Usually, schools uses the square of c (for hypotenuse),square of a (for adjacent or opposite), and square of b (for opposite or adjacent), but the thing is, there is ambiguity sometimes, knowing that square of c can be assigned to one or of the shorter side of a triangle, and a and b to other sides! which is "interesting" to those who are beginners, and still struggling to the theorem. So this page made "initials" to make the formula easy to remember. Take note that we have now and will be using square of h(for hypotenuse), square of a (for adjacent) , and square of o (for opposite). See those initials? no ambiguities. Each letter assigned is the hint itself for what is the particular side of a right triangle we are talking. Recall that right triangle, is a triangle with one angle measure 90 degrees. Take note that we need two known values of sides to use Pythagorean theorem, to solve the unknown. Now, let's try some examples:

Illustrative Examples:

1.) Find the value of the missing side of the given right triangle below.

Based on this figure, what is the missing side? It can be a, b or c, bu if you knew well, it's hypotenuse and c is the missing side. So, by using our Pythagorean formula with initials as we call it, just check the illustration below;

Illustration

${\color{Red} h}^{2}=a^{2}+o^{2}$ , the unknown was highlighted red

${\color{Red} h}^{2}=3^{2}+4^{2}$

${\color{Red} h}^{2}=9+16=25$

${\color{Red} h}=\sqrt{25}=5$

So, we knew now that the missing side of the given triangle above is 5. So the new look of the right triangle is now; see below

2.) Find the value of the missing side of the given right triangle below.

In here, the missing value of a side is different from the previous. It is not the longest, so it is not hypotenuse. By checking where our angle alpha (𝝰) is, we can see what is missing. If you could recall our discussion on Trigonometric Functions of a Right Triangle, we can solve this easily. So the missing side is adjacent based on the position of the angle. Dealing this with our Pythagorean formula below;

$h^{2}={\color{Red} a}^{2}+b^{2}$ , the unknown was highlighted red

$10^{2}={\color{Red}a}^{2}+6^{2}$

${\color{Red}a}^{2}=10^{2}-6^{2}$ , remember Algebra? adding both side by $-6^{2}$ , then flippin' it.

${\color{Red}a}^{2}=100-36$

${\color{Red}a}=\sqrt{64}$

${\color{Red}a}=8$

So we have now the value of the adjacent side of the right triangle. Now, having its missing side, the new right triangle looks like, see below;

3.) Find the value of the missing side of the given right triangle below.

So, finding the value of the missing side which is the opposite side of the right triangle above is by having our modified Pythagorean Theorem formula.

$h^{2}=a^{2}+{\color{Red} o}^{2}$ , this time our missing side is the opposite highlighted in red. To solve, this is by basically substituting the known values of the right triangle;

$11^{2}=7^{2}+{\color{Red} o}^{2}$

$121=49+{\color{Red} o}^{2}$

${\color{Red} o}^{2}=121-49$ , remember Algebra? adding both side by $-49$ ,then flippin' it.

${\color{Red}o}=\sqrt{72}=8.485$ , using calculator would speed up the process.

So the new right triangle of #3 problem is now, see below;

Example Problems with solution:

1.) The right triangle has the longest side measures 96 cm..One of its shorter side measures 50 cm.. what is the length of the other side?

Solution:

We need to illustrate the statement of the problem to make our process more clear. See the illustration below;

Illustration:

So the unknown side is found to be the opposite side. To solve, follow the procedure below.

$h^{2}=a^{2}+{\color{Red} o}^{2}$ , having the Pythagorean formula with unknown highlighted in red

$(96cm.)^{2}=(50cm.)^{2}+{\color{Red} o}^{2}$

${\color{Red} o}^{2}=(96cm.)^{2}-(50cm.)^{2}$ , remember Algebra?

${\color{Red} o}^{2}=9216cm^{2}-2500cm^{2}$

${\color{Red} o} =\sqrt{6716cm^{2}}$

${\color{Red} o} =81.95 cm.$ answer , using calculator would speed up the process.

2.) The two shorter side of a triangle is 5 cm. each. Find its longest side.

Solution: Make an Illustration

See the illustration below.

Illustration.

So the unknown is obviously the hypotenuse. To solve such, let's have this below;

${\color{Red} h}^{2}=a^{2}+o^{2}$ , the unknown was highlighted in red, the hypotenuse

${\color{Red} h}^{2}=(5m.)^{2}+(5m.)^{2}$ , substituting the known values

${\color{Red} h}^{2}=25m^{2}+25m^{2}$

${\color{Red} h}=\sqrt{50m^{2}}$

${\color{Red} h}=7.07m.$ answer , using calculator would speed up the process.

3.) Find the length of the other side of a right triangle whose longest side is 9 inches and one side is 5 inches.

Solution: We should do the illustration first.

Illustration:

Setting up the Pythagorean Formula:

$h^{2}=a^{2}+{\color{Red} o}^{2}$ , or we can apply Algebraically by subtracting $-a^{2}$ both sides, and flip it.

${\color{Red} o}^{2}=h^{2}-a^{2}$ , now the new formula.

${\color{Red} o}^{2}= (9in.)^{2}+(5in.)^{2}$ , substituting what is known.

${\color{Red} o}^{2}= 81in.^{2}+25in.^{2}$

${\color{Red} o}= \sqrt{106in^{2}}$

${\color{Red} o}=10.30in.$ answer , using calculator would speed up the process.

4-5.) Solve the missing side of the given illustration below.

Illustration.

Now, we have to solve for the triangle #1 first. We cannot solve the second triangle ahead because the adjacent side of it, is the opposite side of the first triangle. It means, we need to know the other known value to solve for the hypotenuse side of the second triangle. So,

#4 Solution: for the first triangle, the unknown is opposite, so,

$h^{2}=a^{2}+{\color{Red} o}^{2}$ , or we can apply Algebraically by subtracting $-a^{2}$ both sides, and flip it.

${\color{Red} o}^{2}=h^{2}-a^{2}$ , now the new formula.

${\color{Red} o}^{2}=(5cm.)^{2}-(3cm.)^{2}$ , substituting the known values of the first triangle

${\color{Red} o}^{2}=25cm.^{2}-9cm.^{2}$

${\color{Red} o}=\sqrt{16cm^{2}}$

${\color{Red} o}=4cm.$ , answer (this is the measure of the opposite of the first triangle, and the adjacent of the second)

#5 Solution: for the second triangle, the unknown is hypotenuse, so,

${\color{Red} h}^{2}=a^{2}+o^{2}$ , or simplifying the formula

${\color{Red} h}^{2}=(4cm)^{2}+(6cm)^{2}$

${\color{Red} h}^{2}=16cm^{2}+36cm^{2}$

${\color{Red} h}=\sqrt{52cm^{2}}$

${\color{Red} h}=7.21cm.$ , answer , using calculator would speed up the process.

Now the new triangles with their values looks like, see below;

Practice Test: (With partial solution)

Problem 1-2.) As part of review, the triangle problem below is almost the same as the triangle problem of our Example problems above. The data was intentionally changed but the process is still absolutely the same. Try it.

Illustrated figure:

#1 Solution: for the first triangle, the unknown is opposite, so,

#2 Solution: for the second triangle, the unknown is hypotenuse, so,

${\color{Red} h}^{2}=a^{2}+o^{2}$ , or simplifying the formula

Click here for the answer

Simple Quiz:

1. Find the longest side of a right triangle if one of its side measures 3 meters and the other is 5 meters.

2. If a right triangle has the longest side 50 inches, what is the measure of the other side if one from two of them is half the longest?

3. Warren walks 10 meters to the North direction, turned left, and walks 8 meters. How far did he walked from his original location?

4.) From the figure below, find the hypotenuse of "Triangle 2"