Geometry how to find the missing side of a triangle

Use Pythagoras theorem: "The square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides"

#a^2 + b^2 = 5^2 + 12^2 = 25 + 144 = 169 = 13^2#

So the hypotenuse of your triangle has length #13#.

You have probably encountered the 3-4-5 triangle. The 5-12-13 triangle is part of the same sequence of right angled triangles with integer length sides:

If #k# is a non-negative integer, then

#a=2k+3#

#b=(a^2-1)/2=2k^2+6k+4#

#c=(a^2+1)/2=2k^2+6k+5#

are the lengths of the sides of a right angled triangle.

How to Find the Missing Side Length Given Two Similar Triangles

Step 1: Given a pair of similar triangles, determine which sides correspond to each other. The smallest side of the first triangle corresponds to the smallest side of the second triangle. The largest side of the first triangle corresponds to the largest side of the second triangle. And, of course, the remaining side of the first triangle corresponds to the remaining side of the second triangle. Once we've established the correct correspondence, say {eq}\triangle ABC\sim\triangle DEF {/eq}, identify two corresponding sides whose side lengths are known, say {eq}{AB} {/eq} and {eq}{DE} {/eq}, and then form the ratio {eq}\dfrac{ {AB}}{ {DE}} {/eq} of the side lengths.

Step 2: Next, form the ratio of corresponding side lengths where one of the side lengths involves the unknown, and set this ratio equal to the ratio found in Step 1.

Step 3: Cross multiply and then solve for the missing side length.

How to Find the Missing Side Length Given Two Similar Triangles: Vocabulary

Similar triangles: Triangles {eq}\triangle ABC {/eq} and {eq}\triangle DEF {/eq} are similar, written {eq}\triangle ABC \sim\triangle DEF {/eq}, if the two triangles are the same shape but different sizes. We say {eq}{AB} {/eq} corresponds to {eq}{DE} {/eq}, {eq}{BC} {/eq} corresponds to {eq}{EF} {/eq}, and {eq}{CA} {/eq} corresponds to {eq}{FD} {/eq}. More precisely, {eq}\triangle ABC \sim\triangle DEF {/eq} if the ratios of the corresponding sides are equal:

$$\dfrac{ {AB}}{ {DE}}=\dfrac{ {BC}}{ {EF}}=\dfrac{ {CA}}{ {FD}} $$

Scale factor: For a pair of similar triangles, the scale factor is the common ratio of corresponding side lengths, but in reduced form. The scale factor can be written as {eq}\frac{a}{b} {/eq} or {eq}a:b {/eq}.

Now let's practice finding missing side lengths given two similar triangles with two typical examples.

How to Find the Missing Side Length Given Two Similar Triangles: Example 1

{eq}\triangle ABC {/eq} and {eq}\triangle RST {/eq} have side lengths as indicated in the figure below. Given that the two triangles are similar, find the missing side length {eq}x {/eq}.

Step 1: Judging by the relative lengths of the sides of the two triangles, we can conclude that A corresponds to T, B to S, and C to R, which we can write as {eq}\triangle ABC\sim \triangle TSR. {/eq} In particular, the smallest sides {eq}AB {/eq} and {eq}TS {/eq} are corresponding sides with known lengths, so we form the ratio of their side lengths {eq}\dfrac{9}{15} {/eq}. Remember that all ratios of corresponding side lengths are equal.

Step 2: Next we form a ratio that involves {eq}x {/eq}. Since {eq}BC {/eq} and {eq}SR {/eq} are corresponding sides, we know that the ratio of their side lengths equals the ratio from Step 1: {eq}\dfrac{18}{x}=\dfrac{9}{15} {/eq}

Step 3: Finally, we cross-multiply and solve for {eq}x {/eq}:

{eq}\begin{align} x\cdot 9&=18\cdot 15 \\ 9x&=270 \\ x&=\dfrac{270}{9} \\ &=30 \end{align} {/eq}

That is, the missing side length is 30.

How to Find the Missing Side Length Given Two Similar Triangles: Example 2

The two triangles pictured below are similar. Find the missing side length {eq}x {/eq}.

Step 1: Looking at the relative lengths of the sides, we see that {eq}\triangle ABC\sim \triangle EDF {/eq}. In particular, the shortest sides {eq}AC=15 {/eq} and {eq}EF=6 {/eq} are corresponding sides. The ratio of these side lengths is {eq}\dfrac{15}{6} {/eq}

Step 2: The corresponding sides that involve {eq}x {/eq} are {eq}AB=30 {/eq} and {eq}ED=x {/eq}, so we form the ratio of them and set it equal to the ratio found in Step 1: {eq}\dfrac{30}{x}=\dfrac{15}{6} {/eq}

Step 3: Cross-multiplying and solving, we get:

{eq}\begin{align} 15x&=180 \\ x&=12 \end{align} {/eq}

The missing side length is 12.

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