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Maths-

General

Easy

Question

Which of the following additional measurements to the triangle provides enough information to determine the length of $stack A B with minus on top$ ?
I. The length of $stack A C with minus on top$
II. The measure of $angle A B C$

I only
II only
Either I or II
Neither I nor II

The correct answer is: Either I or II

For a right-angled triangle, $cos invisible function application angle equals fraction numerator b a s e over denominator h y p o t e n u s e end fraction$ and by Pythagoras’ theorem, $open parentheses h y p o t e n u s e close parentheses to the power of 2 end exponent equals open parentheses b a s e close parentheses to the power of 2 end exponent plus open parentheses h e i g h t close parentheses to the power of 2 end exponent$
Explanations:
Step 1 of 2:
If “I. The length of $stack A C with minus on top$ ” is provided, then by Pythagoras’ theorem, we could find
$open parentheses h e i g h t close parentheses to the power of 2 end exponent equals open parentheses h y p o t e n u s e close parentheses to the power of 2 end exponent minus open parentheses b a s e close parentheses to the power of 2 end exponent$
$rightwards double arrow stack A B with minus on top to the power of 2 end exponent equals stack B C with minus on top to the power of 2 end exponent minus stack A C with minus on top to the power of 2 end exponent$
$rightwards double arrow stack A B with minus on top equals square root of 25 to the power of 2 end exponent minus stack A C with minus on top to the power of 2 end exponent end root$ , since it is a distance, we consider only the positive sq. root value of $stack A B with minus on top$ .
So, the length of $stack A C with minus on top$ would be enough information to determine the length of $stack A B with minus on top$ .
Step 2 of 2:
If “II. The measure of $angle A B C$ ” is provided, then w.r.t. $angle B$ , $stack A B with minus on top$ is the base and $stack B C with minus on top equals 25$ is the hypotenuse.
So, $cos invisible function application B equals fraction numerator stack A B with minus on top over denominator stack B C with minus on top end fraction equals fraction numerator stack A B with minus on top over denominator 25 end fraction$
$rightwards double arrow stack A B with minus on top equals cos invisible function application B cross times 25$
As $angle A B C equals angle B$ is known then $stack A B with minus on top$ can be determined.
So, this also satisfies.
Final Answer:
The right option is— Either I or II.

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