Triangles - ACT Math
Card 1 of 30
What is the formula for the perimeter of a triangle?
What is the formula for the perimeter of a triangle?
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Sum of all side lengths. Add all three side lengths: $a + b + c$.
Sum of all side lengths. Add all three side lengths: $a + b + c$.
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Find the area of a right triangle with legs 6 and 8.
Find the area of a right triangle with legs 6 and 8.
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$24$ square units. Using $\frac{1}{2} \times 6 \times 8 = 24$ for right triangle legs.
$24$ square units. Using $\frac{1}{2} \times 6 \times 8 = 24$ for right triangle legs.
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Calculate the perimeter of a triangle with sides 3, 4, 5.
Calculate the perimeter of a triangle with sides 3, 4, 5.
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- Sum of all three sides: $3 + 4 + 5 = 12$.
- Sum of all three sides: $3 + 4 + 5 = 12$.
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What is an altitude in a triangle?
What is an altitude in a triangle?
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A perpendicular segment from a vertex to the opposite side. Creates the shortest distance from vertex to opposite side.
A perpendicular segment from a vertex to the opposite side. Creates the shortest distance from vertex to opposite side.
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Which triangle inequality must hold for any triangle?
Which triangle inequality must hold for any triangle?
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The sum of any two sides must be greater than the third side. Ensures triangle can actually be formed from three sides.
The sum of any two sides must be greater than the third side. Ensures triangle can actually be formed from three sides.
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Identify the type of triangle with all sides equal.
Identify the type of triangle with all sides equal.
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Equilateral triangle. All three sides are identical in length.
Equilateral triangle. All three sides are identical in length.
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What is a median in a triangle?
What is a median in a triangle?
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A segment from a vertex to the midpoint of the opposite side. Connects vertex to midpoint of opposite side.
A segment from a vertex to the midpoint of the opposite side. Connects vertex to midpoint of opposite side.
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What is the relationship between an exterior angle and its remote interior angles?
What is the relationship between an exterior angle and its remote interior angles?
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Exterior angle = sum of remote interior angles. Exterior angle equals sum of the two non-adjacent interior angles.
Exterior angle = sum of remote interior angles. Exterior angle equals sum of the two non-adjacent interior angles.
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Find the hypotenuse in a 30-60-90 triangle with short leg 5.
Find the hypotenuse in a 30-60-90 triangle with short leg 5.
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- In 30-60-90 triangle, hypotenuse is twice the short leg.
- In 30-60-90 triangle, hypotenuse is twice the short leg.
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What is the measure of each angle in an equilateral triangle?
What is the measure of each angle in an equilateral triangle?
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60 degrees. Since all sides are equal, all angles are equal: $180° ÷ 3 = 60°$.
60 degrees. Since all sides are equal, all angles are equal: $180° ÷ 3 = 60°$.
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What do the exterior angles of a triangle sum to?
What do the exterior angles of a triangle sum to?
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360°. Each exterior angle plus its interior angle equals $180°$.
360°. Each exterior angle plus its interior angle equals $180°$.
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What is the sum of the interior angles of any triangle in degrees?
What is the sum of the interior angles of any triangle in degrees?
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$180^\circ$. The angles in any triangle always add up to this value.
$180^\circ$. The angles in any triangle always add up to this value.
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State the Cosine Rule for triangle sides $a$, $b$, $c$ and angle $C$.
State the Cosine Rule for triangle sides $a$, $b$, $c$ and angle $C$.
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$c^2 = a^2 + b^2 - 2ab \cos C$. General formula connecting all sides and one angle.
$c^2 = a^2 + b^2 - 2ab \cos C$. General formula connecting all sides and one angle.
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Which triangle inequality must hold for any triangle?
Which triangle inequality must hold for any triangle?
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The sum of any two sides must be greater than the third side. Ensures triangle can actually be formed from three sides.
The sum of any two sides must be greater than the third side. Ensures triangle can actually be formed from three sides.
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What is the circumcenter of a triangle?
What is the circumcenter of a triangle?
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The point where the perpendicular bisectors of the sides meet. Center of the circle passing through all three vertices.
The point where the perpendicular bisectors of the sides meet. Center of the circle passing through all three vertices.
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Identify the type of triangle with angles 90°, 45°, 45°.
Identify the type of triangle with angles 90°, 45°, 45°.
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Isosceles right triangle. Right triangle with two equal legs and equal base angles.
Isosceles right triangle. Right triangle with two equal legs and equal base angles.
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What is the orthocenter of a triangle?
What is the orthocenter of a triangle?
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The point where the altitudes meet. Intersection point of all three altitude lines.
The point where the altitudes meet. Intersection point of all three altitude lines.
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State the formula for the area of an equilateral triangle with side $s$.
State the formula for the area of an equilateral triangle with side $s$.
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$\frac{\sqrt{3}}{4} s^2$. Derived from base-height formula using equilateral properties.
$\frac{\sqrt{3}}{4} s^2$. Derived from base-height formula using equilateral properties.
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Given a triangle with sides 7, 24, 25, classify it.
Given a triangle with sides 7, 24, 25, classify it.
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Right triangle. Check: $7^2 + 24^2 = 49 + 576 = 625 = 25^2$, so right triangle.
Right triangle. Check: $7^2 + 24^2 = 49 + 576 = 625 = 25^2$, so right triangle.
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What is the angle opposite the largest side in a triangle?
What is the angle opposite the largest side in a triangle?
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Largest angle. In any triangle, the largest angle faces the longest side.
Largest angle. In any triangle, the largest angle faces the longest side.
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Find the third angle in a triangle with angles 35° and 65°.
Find the third angle in a triangle with angles 35° and 65°.
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80°. Sum of angles is $180°$: $180° - 35° - 65° = 80°$.
80°. Sum of angles is $180°$: $180° - 35° - 65° = 80°$.
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Identify the type of triangle with sides 5, 5, and 8.
Identify the type of triangle with sides 5, 5, and 8.
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Isosceles triangle. Two sides are equal ($5$ each), third side different.
Isosceles triangle. Two sides are equal ($5$ each), third side different.
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What is the semi-perimeter of a triangle with sides $a$, $b$, $c$?
What is the semi-perimeter of a triangle with sides $a$, $b$, $c$?
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$\frac{a+b+c}{2}$. Half the total perimeter, used in area formulas.
$\frac{a+b+c}{2}$. Half the total perimeter, used in area formulas.
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What is the incenter of a triangle?
What is the incenter of a triangle?
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The point where the angle bisectors meet. Center of the circle inscribed within the triangle.
The point where the angle bisectors meet. Center of the circle inscribed within the triangle.
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What is the centroid of a triangle?
What is the centroid of a triangle?
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The point where the medians meet. Balance point where triangle would rest on a pin.
The point where the medians meet. Balance point where triangle would rest on a pin.
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Find the missing angle if two angles are 45° and 60°.
Find the missing angle if two angles are 45° and 60°.
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75°. Sum of angles is $180°$: $180° - 45° - 60° = 75°$.
75°. Sum of angles is $180°$: $180° - 45° - 60° = 75°$.
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Define an isosceles triangle.
Define an isosceles triangle.
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A triangle with two equal sides. Two sides have equal length, creating two equal angles.
A triangle with two equal sides. Two sides have equal length, creating two equal angles.
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Which triangle has one angle greater than 90°?
Which triangle has one angle greater than 90°?
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Obtuse triangle. One angle exceeds $90°$, making it wider than a right angle.
Obtuse triangle. One angle exceeds $90°$, making it wider than a right angle.
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What is the altitude in a triangle?
What is the altitude in a triangle?
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A perpendicular segment from a vertex to the opposite side. Height drops perpendicularly from vertex to opposite side.
A perpendicular segment from a vertex to the opposite side. Height drops perpendicularly from vertex to opposite side.
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Calculate the area of a triangle with base 10 and height 5.
Calculate the area of a triangle with base 10 and height 5.
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25 square units. Using $\frac{1}{2} \times 10 \times 5 = 25$.
25 square units. Using $\frac{1}{2} \times 10 \times 5 = 25$.
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