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📝 Right triangle properties (12 MCQs)

📖 From Digital SAT Algebra • 3. Mathematical Models in Algebra • 12 questions available

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What is Right triangle properties?

Definition:
A right triangle is a triangle that has one angle exactly $90^\circ$ , and its properties include the Pythagorean theorem ( $a^2 + b^2 = c^2$ ), where $c$ is the hypotenuse (the longest side opposite the right angle), the trigonometric ratios (sine, cosine, tangent), and the fact that the other two angles are acute and sum to $90^\circ$ , making them complementary.

Working:
The Pythagorean theorem is used to find missing side lengths, where the hypotenuse is found by $c = \sqrt{a^2 + b^2}$ , and a leg by $a = \sqrt{c^2 - b^2}$ , and the trigonometric functions relate the angles to side ratios, such as $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ , $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$ , and $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ .

Example:
A right triangle has legs of 5 and 12, then the hypotenuse is $c = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$ , and the angles can be found using $\sin(\theta) = \frac{5}{13}$ or $\cos(\theta) = \frac{12}{13}$ , demonstrating the key properties.

Reason:
Right triangle properties are essential in geometry, trigonometry, physics, and engineering for solving problems involving distances, heights, slopes, and vector components, and they are the basis of many measurements and calculations in real-world applications.

Easy

Medium

Hard

📝 All Right triangle properties MCQs

Q1. Which statement best identifies a right triangle when examining its angles?

A.It has exactly one angle measuring

90^\circ

✅

B.It must have two equal acute angles

C.All three angles must be less than

90^\circ

D.Its largest angle must measure

180^\circ

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: A right triangle is defined by having exactly one $90^\circ$ angle. The other two angles must be acute and together measure $90^\circ$ . Two equal acute angles may occur, but they are not required.

Q2. A triangle has angles $38^\circ$ , $52^\circ$ , and $90^\circ$ . A student says it cannot be a right triangle because the acute angles are unequal. What is the best evaluation?

A.The student is correct because right triangles require equal acute angles

B.The student is incorrect because only one angle must be

90^\circ

✅

C.The student is correct because all angles must be equal

D.The student is incorrect because a right triangle has two

90^\circ

angles

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The student's reasoning confuses a right triangle with a special right triangle. A right triangle only requires one $90^\circ$ angle. The other two angles can have different measures, as $38^\circ+52^\circ=90^\circ$ .

Q3. A triangular support has angles $90^\circ$ , $35^\circ$ , and $55^\circ$ . If the support is redesigned so that the $35^\circ$ angle becomes $40^\circ$ , what must happen to the remaining acute angle?

A.It becomes

45^\circ

B.It becomes

50^\circ

✅

C.It becomes

55^\circ

D.It becomes

60^\circ

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The two acute angles in a right triangle must have a combined measure of $90^\circ$ . Increasing one from $35^\circ$ to $40^\circ$ forces the other to decrease from $55^\circ$ to $50^\circ$ .

Q4. A student draws a triangle with angles $90^\circ$ , $90^\circ$ , and $0^\circ$ and calls it a right triangle. Which reasoning most accurately identifies the error?

A.The angles add to

180^\circ

, but a triangle cannot have two right angles ✅

B.A right triangle must have three right angles

C.The zero-degree angle should be

90^\circ

D.The total angle measure of a triangle must be

360^\circ

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Although the listed angles add to $180^\circ$ , two $90^\circ$ angles leave no positive angle for the third vertex. A valid triangle can have only one right angle because its other two angles must be positive acute angles.

Q5. A carpenter needs a triangular brace with one corner fixed at $90^\circ$ . The other two corners are planned as $2x^\circ$ and $x^\circ$ . Which value of $x$ produces a valid right triangle?

A.20

B.30 ✅

C.45

D.60

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The angles of the triangle must total $180^\circ$ . Therefore, $90+2x+x=180$ , giving $3x=90$ and $x=30$ . The resulting angles are $90^\circ$ , $60^\circ$ , and $30^\circ$ , which form a valid right triangle.

Q6. A triangular ramp is modeled with angles $90^\circ$ , $28^\circ$ , and $62^\circ$ . The designer wants to increase the smaller acute angle by $7^\circ$ while keeping the triangle right-angled. What should the other acute angle become?

A.55°

B.62°

C.65° ✅

D.69°

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The smaller acute angle changes from $28^\circ$ to $35^\circ$ . Because the triangle remains right-angled, its two acute angles must sum to $90^\circ$ , so the other angle must be $90-35=55^\circ$ .

Q7. A coordinate graph shows three points $A(1,1)$ , $B(1,6)$ , and $C(5,1)$ . What feature of the plotted segments allows you to identify the triangle as right-angled?

A.The two legs have the same length

B.The segments from

A

B

and

A

C

are perpendicular ✅

C.All three points have positive coordinates

D.The segment

BC

is horizontal

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Segment $AB$ is vertical because the $x$ -coordinate is constant, while $AC$ is horizontal because the $y$ -coordinate is constant. A vertical and horizontal segment meeting at $A$ form a $90^\circ$ angle.

Q8. A student claims that any triangle containing an angle larger than $90^\circ$ is also a right triangle because it has a very large angle. Why is this reasoning incorrect?

A.A right triangle must contain exactly one

90^\circ

angle, not an obtuse angle ✅

B.A triangle can never contain an angle larger than

90^\circ

C.Right triangles must contain two obtuse angles

D.The largest angle in every triangle is always

60^\circ

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: An angle larger than $90^\circ$ is obtuse, so a triangle containing one is classified as obtuse rather than right. A right triangle specifically has one angle equal to $90^\circ$ , with the other two angles acute.

Q9. A triangular window has one angle of $90^\circ$ . One installer measures another angle as $47^\circ$ , while a second installer measures it as $43^\circ$ . Assuming one measurement is correct, which conclusion follows?

A.The triangle cannot be right-angled

B.The third angle must be

90^\circ

C.The third angle is either

43^\circ

47^\circ

depending on which measurement is correct ✅

D.Both measurements must be correct

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: If the measured acute angle is $47^\circ$ , the other acute angle is $43^\circ$ . If it is $43^\circ$ , the other is $47^\circ$ . Thus the third angle can be $43^\circ$ or $47^\circ$ , while the known right angle remains $90^\circ$ .

Q10. A student reasons: $90^\circ+48^\circ+48^\circ=186^\circ$ , so a triangle with one right angle and two equal angles of $48^\circ$ is possible because the two acute angles are close to $45^\circ$ . What is the precise error?

A.The right angle should be

80^\circ

B.The acute angles should each be

45^\circ

because they must sum to

90^\circ

✅

C.The acute angles should each be

48^\circ

but the total should be

190^\circ

D.A triangle may have any total angle measure

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Once one angle is $90^\circ$ , the other two angles together must equal $90^\circ$ . If they are equal, each must be $45^\circ$ . The proposed $48^\circ$ angles produce a total of $186^\circ$ , so the model is impossible.

Q11. A triangular frame has one angle fixed at $90^\circ$ . Its two other angles are in the ratio $2:3$ . Which pair of angles completes the frame?

A.30° and 60°

B.32° and 58°

C.36° and 54° ✅

D.40° and 50°

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The two acute angles must total $90^\circ$ . Let them be $2x$ and $3x$ . Then $5x=90$ , so $x=18$ . Therefore the angles are $36^\circ$ and $54^\circ$ , satisfying both the ratio and right-triangle conditions.

Q12. A designer compares two triangular supports. Support P has angles $90^\circ,20^\circ,70^\circ$ , while Support Q has angles $90^\circ,45^\circ,45^\circ$ . Which conclusion is most accurate?

A.Only P is a right triangle

B.Only Q is a right triangle

C.Both are right triangles, but their acute-angle structures differ ✅

D.Neither is a right triangle because the angles are unequal

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Both supports contain exactly one $90^\circ$ angle, so both are right triangles. Support P has unequal acute angles of $20^\circ$ and $70^\circ$ , whereas Support Q has equal acute angles of $45^\circ$ each.

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