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It is the most powerful prayer. A pure heart, a clean mind, and a clear conscience is necessary for it.
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For in GOD we live, and move, and have our being. - Acts 17:28

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# Solved Examples on Congruent Triangles and Similar Triangles

For ACT Students
The ACT is a timed exam...$60$ questions for $60$ minutes
This implies that you have to solve each question in one minute.
Some questions will typically take less than a minute a solve.
Some questions will typically take more than a minute to solve.
The goal is to maximize your time. You use the time saved on those questions you solved in less than a minute, to solve the questions that will take more than a minute.
So, you should try to solve each question correctly and timely.
So, it is not just solving a question correctly, but solving it correctly on time.
Please ensure you attempt all ACT questions.
There is no "negative" penalty for any wrong answer.

For JAMB Students
Calculators are not allowed. So, the questions are solved in a way that does not require a calculator.

For WASSCE Students
Any question labeled WASCCE is a question for the WASCCE General Mathematics
Any question labeled WASSCE:FM is a question for the WASSCE Further Mathematics/Elective Mathematics

For GCSE and Malta Students
All work is shown to satisfy (and actually exceed) the minimum for awarding method marks.
Calculators are allowed for some questions. Calculators are not allowed for some questions.

For NSC Students
For the Questions:
Any space included in a number indicates a comma used to separate digits...separating multiples of three digits from behind.
Any comma included in a number indicates a decimal point.
For the Solutions:
Decimals are used appropriately rather than commas
Commas are used to separate digits appropriately.

Solve all questions
State the reasons for applicable steps
Show all work

(1.) CSEC

Triangles PQR and STR are similar triangles.
(i) Complete the following statement:
In the diagram above, the corresponding angles of $\triangle$ PQR and $\triangle$ STR are ..................... and the ......................... of their corresponding sides are the same.

In the diagram above, not drawn to scale, RS = 15 cm, SP = 9 cm and ST = 12 cm.
(ii) Determine the length of PQ

(i) Complete the following statement:
In the diagram above, the corresponding angles of $\triangle$ PQR and $\triangle$ STR are equal and the ratio of their corresponding sides are the same.

$\triangle PQR \sim \triangle STR \\[3ex] \dfrac{12}{|PQ|} = \dfrac{15}{15 + 9} \\[5ex] \dfrac{12}{|PQ|} = \dfrac{15}{24} \\[5ex] 15 * |PQ| = 12 * 24 \\[3ex] |PQ| = \dfrac{12 * 24}{15} \\[5ex] |PQ| = \dfrac{288}{15} \\[5ex] |PQ| = 19.2\;cm$
(2.) ACT The lengths of the sides of a triangle are 3, 8, and 9 inches.
How many inches long is the shortest side of a similar triangle that has a perimeter of 60 inches?

$F.\;\; 9 \\[3ex] G.\;\; 11 \\[3ex] H.\;\; 20 \\[3ex] J.\;\; 24 \\[3ex] K.\;\; 27 \\[3ex]$

$Perimeter\;\;of\;\;1st\;\;\triangle = 3 + 8 + 9 = 20\;inches \\[3ex] Perimeter\;\;of\;\;similar\;\;\triangle = 60\;inches \\[3ex] Shortest\;\;side\;\;of\;\;1st\;\;\triangle = 3\;inches \\[3ex] Shortest\;\;side\;\;of\;\;similar\;\;\triangle = p\;inches \\[3ex] \dfrac{20}{60} = \dfrac{3}{p}...Perimeter\;\;Ratio = Scale\;\;Factor \\[5ex] \dfrac{1}{3} = \dfrac{3}{p} \\[5ex] 1 * p = 3 * 3 \\[3ex] p = 9\;inches$
(3.) GCSE Here are two right-angled triangles.

Circle the value of y
11        7.5        9        4

Because two of the angles of both right triangles: 90° and x° are the same (regardless of their different side lengths), the two triangles are similar...AA Similarity Postulate

$\dfrac{y}{15} = \dfrac{6}{10}...Scale\;\;Factor \\[5ex] y = \dfrac{15 * 6}{10} \\[5ex] y = \dfrac{15 * 3}{5} \\[5ex] y = 3(3) \\[3ex] y = 9$
(4.)


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(6.)