Trigonometric Ratios from Right Triangle Similarity - Geometry
Card 1 of 30
What does it mean for two triangles to be congruent in terms of corresponding parts?
What does it mean for two triangles to be congruent in terms of corresponding parts?
Tap to reveal answer
$\text{All corresponding sides and angles are equal}$. Congruent triangles are identical in size and shape.
$\text{All corresponding sides and angles are equal}$. Congruent triangles are identical in size and shape.
← Didn't Know|Knew It →
Find $x$ if $\triangle ABC\sim\triangle DEF$, $\frac{AB}{DE}=\frac{3}{7}$, and $AB=12$ with $DE=x$.
Find $x$ if $\triangle ABC\sim\triangle DEF$, $\frac{AB}{DE}=\frac{3}{7}$, and $AB=12$ with $DE=x$.
Tap to reveal answer
$x=12\cdot\frac{7}{3}=28$. If $\frac{AB}{DE} = \frac{3}{7}$, then $DE = AB \cdot \frac{7}{3}$.
$x=12\cdot\frac{7}{3}=28$. If $\frac{AB}{DE} = \frac{3}{7}$, then $DE = AB \cdot \frac{7}{3}$.
← Didn't Know|Knew It →
If $\triangle ABC\sim\triangle DEF$, which angle corresponds to $\angle C$?
If $\triangle ABC\sim\triangle DEF$, which angle corresponds to $\angle C$?
Tap to reveal answer
$\angle F$. Third angles in similarity statement correspond.
$\angle F$. Third angles in similarity statement correspond.
← Didn't Know|Knew It →
If $\triangle ABC\sim\triangle DEF$, which side corresponds to $AB$?
If $\triangle ABC\sim\triangle DEF$, which side corresponds to $AB$?
Tap to reveal answer
$DE$. First sides in similarity statement correspond.
$DE$. First sides in similarity statement correspond.
← Didn't Know|Knew It →
What is the relationship between the scale factor $k$ of similar triangles and their areas?
What is the relationship between the scale factor $k$ of similar triangles and their areas?
Tap to reveal answer
$\frac{A_1}{A_2}=k^2$. Areas scale with the square of the scale factor.
$\frac{A_1}{A_2}=k^2$. Areas scale with the square of the scale factor.
← Didn't Know|Knew It →
If $\triangle ABC\sim\triangle DEF$, which side corresponds to $AB$?
If $\triangle ABC\sim\triangle DEF$, which side corresponds to $AB$?
Tap to reveal answer
$DE$. First sides in similarity statement correspond.
$DE$. First sides in similarity statement correspond.
← Didn't Know|Knew It →
What similarity criterion states that if two angles are congruent, then the triangles are similar?
What similarity criterion states that if two angles are congruent, then the triangles are similar?
Tap to reveal answer
$\text{AA similarity}$. Two angles determine the third, making triangles similar.
$\text{AA similarity}$. Two angles determine the third, making triangles similar.
← Didn't Know|Knew It →
Which statement is always true if $\triangle ABC\cong\triangle DEF$: $\frac{AB}{DE}=1$ or $\frac{AB}{DE}=k$ for any $k$?
Which statement is always true if $\triangle ABC\cong\triangle DEF$: $\frac{AB}{DE}=1$ or $\frac{AB}{DE}=k$ for any $k$?
Tap to reveal answer
$\frac{AB}{DE}=1$. Congruent triangles have scale factor 1 (equal sides).
$\frac{AB}{DE}=1$. Congruent triangles have scale factor 1 (equal sides).
← Didn't Know|Knew It →
Which criterion guarantees triangle congruence when $\angle A\cong\angle D$, $\angle B\cong\angle E$, and $\angle C\cong\angle F$?
Which criterion guarantees triangle congruence when $\angle A\cong\angle D$, $\angle B\cong\angle E$, and $\angle C\cong\angle F$?
Tap to reveal answer
$\text{None; AAA gives similarity, not congruence}$. AAA proves similarity only, not congruence.
$\text{None; AAA gives similarity, not congruence}$. AAA proves similarity only, not congruence.
← Didn't Know|Knew It →
Find $x$ if $\triangle ABC\sim\triangle DEF$, $AB=6$ corresponds to $DE=15$, and $\text{area}(ABC)=24$ with $\text{area}(DEF)=x$.
Find $x$ if $\triangle ABC\sim\triangle DEF$, $AB=6$ corresponds to $DE=15$, and $\text{area}(ABC)=24$ with $\text{area}(DEF)=x$.
Tap to reveal answer
$x=24\cdot\left(\frac{15}{6}\right)^2=150$. Areas scale with square of side ratio: $\left(\frac{15}{6}\right)^2$.
$x=24\cdot\left(\frac{15}{6}\right)^2=150$. Areas scale with square of side ratio: $\left(\frac{15}{6}\right)^2$.
← Didn't Know|Knew It →
Find $x$ if $\triangle ABC\sim\triangle DEF$, $AB=12$ corresponds to $DE=9$, and $\text{perimeter}(ABC)=40$ with $\text{perimeter}(DEF)=x$.
Find $x$ if $\triangle ABC\sim\triangle DEF$, $AB=12$ corresponds to $DE=9$, and $\text{perimeter}(ABC)=40$ with $\text{perimeter}(DEF)=x$.
Tap to reveal answer
$x=40\cdot\frac{9}{12}=30$. Perimeters scale with same factor as sides: $\frac{9}{12}$.
$x=40\cdot\frac{9}{12}=30$. Perimeters scale with same factor as sides: $\frac{9}{12}$.
← Didn't Know|Knew It →
Find $x$ if $\triangle ABC\sim\triangle DEF$, $AB=5$, $AC=7$, $DE=15$, and $DF=x$ corresponds to $AC$.
Find $x$ if $\triangle ABC\sim\triangle DEF$, $AB=5$, $AC=7$, $DE=15$, and $DF=x$ corresponds to $AC$.
Tap to reveal answer
$x=7\cdot\frac{15}{5}=21$. Scale factor from ABC to DEF is $\frac{15}{5} = 3$.
$x=7\cdot\frac{15}{5}=21$. Scale factor from ABC to DEF is $\frac{15}{5} = 3$.
← Didn't Know|Knew It →
Identify the criterion: Two right triangles have equal hypotenuse and one equal leg.
Identify the criterion: Two right triangles have equal hypotenuse and one equal leg.
Tap to reveal answer
$\text{HL congruence}$. Hypotenuse-Leg is specific to right triangles.
$\text{HL congruence}$. Hypotenuse-Leg is specific to right triangles.
← Didn't Know|Knew It →
Identify the criterion: In right triangles, $\angle C=\angle F=90^\circ$, $AC=DF$, and $BC=EF$.
Identify the criterion: In right triangles, $\angle C=\angle F=90^\circ$, $AC=DF$, and $BC=EF$.
Tap to reveal answer
$\text{SAS congruence}$. Two sides and included right angle match SAS.
$\text{SAS congruence}$. Two sides and included right angle match SAS.
← Didn't Know|Knew It →
What conclusion follows if $\triangle ABC\sim\triangle DEF$ regarding $\frac{AB}{DE}$ and $\frac{BC}{EF}$?
What conclusion follows if $\triangle ABC\sim\triangle DEF$ regarding $\frac{AB}{DE}$ and $\frac{BC}{EF}$?
Tap to reveal answer
$\frac{AB}{DE}=\frac{BC}{EF}$. Similar triangles have all corresponding side ratios equal.
$\frac{AB}{DE}=\frac{BC}{EF}$. Similar triangles have all corresponding side ratios equal.
← Didn't Know|Knew It →
What conclusion follows if $\triangle ABC\sim\triangle DEF$ regarding $\angle A$ and $\angle D$?
What conclusion follows if $\triangle ABC\sim\triangle DEF$ regarding $\angle A$ and $\angle D$?
Tap to reveal answer
$\angle A\cong\angle D$. Similar triangles have all corresponding angles congruent.
$\angle A\cong\angle D$. Similar triangles have all corresponding angles congruent.
← Didn't Know|Knew It →
Find $BC$ if $\triangle ABC\sim\triangle DEF$, $k=\frac{3}{2}$ from $ABC$ to $DEF$, and $EF=12$ corresponds to $BC$.
Find $BC$ if $\triangle ABC\sim\triangle DEF$, $k=\frac{3}{2}$ from $ABC$ to $DEF$, and $EF=12$ corresponds to $BC$.
Tap to reveal answer
$BC=12\cdot\frac{2}{3}=8$. Reverse the scale factor: $BC = EF \cdot \frac{2}{3}$.
$BC=12\cdot\frac{2}{3}=8$. Reverse the scale factor: $BC = EF \cdot \frac{2}{3}$.
← Didn't Know|Knew It →
Find $EF$ if $\triangle ABC\sim\triangle DEF$, $k=\frac{3}{2}$ from $ABC$ to $DEF$, and $BC=8$ corresponds to $EF$.
Find $EF$ if $\triangle ABC\sim\triangle DEF$, $k=\frac{3}{2}$ from $ABC$ to $DEF$, and $BC=8$ corresponds to $EF$.
Tap to reveal answer
$EF=8\cdot\frac{3}{2}=12$. Apply scale factor: $EF = BC \cdot k$.
$EF=8\cdot\frac{3}{2}=12$. Apply scale factor: $EF = BC \cdot k$.
← Didn't Know|Knew It →
Find the scale factor $k$ (from $\triangle ABC$ to $\triangle DEF$) if $BC=11$ corresponds to $EF=22$.
Find the scale factor $k$ (from $\triangle ABC$ to $\triangle DEF$) if $BC=11$ corresponds to $EF=22$.
Tap to reveal answer
$k=\frac{22}{11}=2$. Scale factor is ratio of corresponding sides.
$k=\frac{22}{11}=2$. Scale factor is ratio of corresponding sides.
← Didn't Know|Knew It →
Which criterion proves similarity when you know only that $\angle A\cong\angle D$ and $\angle B\cong\angle E$?
Which criterion proves similarity when you know only that $\angle A\cong\angle D$ and $\angle B\cong\angle E$?
Tap to reveal answer
$\text{AA similarity}$. Two congruent angles are sufficient for AA similarity.
$\text{AA similarity}$. Two congruent angles are sufficient for AA similarity.
← Didn't Know|Knew It →
Which triangle criterion proves similarity when two sides are in ratio $\frac{5}{8}$ and $\frac{10}{16}$ and the included angle is equal?
Which triangle criterion proves similarity when two sides are in ratio $\frac{5}{8}$ and $\frac{10}{16}$ and the included angle is equal?
Tap to reveal answer
$\text{SAS similarity}$. Ratios $\frac{5}{8} = \frac{10}{16}$ with included angle match SAS.
$\text{SAS similarity}$. Ratios $\frac{5}{8} = \frac{10}{16}$ with included angle match SAS.
← Didn't Know|Knew It →
Find $x$ if $\triangle ABC\sim\triangle DEF$, $\frac{AB}{DE}=\frac{3}{7}$, and $AB=12$ with $DE=x$.
Find $x$ if $\triangle ABC\sim\triangle DEF$, $\frac{AB}{DE}=\frac{3}{7}$, and $AB=12$ with $DE=x$.
Tap to reveal answer
$x=12\cdot\frac{7}{3}=28$. If $\frac{AB}{DE} = \frac{3}{7}$, then $DE = AB \cdot \frac{7}{3}$.
$x=12\cdot\frac{7}{3}=28$. If $\frac{AB}{DE} = \frac{3}{7}$, then $DE = AB \cdot \frac{7}{3}$.
← Didn't Know|Knew It →
Find $x$ if $\triangle ABC\sim\triangle DEF$, $AB=4$, $BC=6$, $DE=10$, and $EF=x$ corresponds to $BC$.
Find $x$ if $\triangle ABC\sim\triangle DEF$, $AB=4$, $BC=6$, $DE=10$, and $EF=x$ corresponds to $BC$.
Tap to reveal answer
$x=6\cdot\frac{10}{4}=15$. Scale factor from ABC to DEF is $\frac{10}{4} = \frac{5}{2}$.
$x=6\cdot\frac{10}{4}=15$. Scale factor from ABC to DEF is $\frac{10}{4} = \frac{5}{2}$.
← Didn't Know|Knew It →
What does it mean for sides to be proportional in similar triangles using a scale factor $k$?
What does it mean for sides to be proportional in similar triangles using a scale factor $k$?
Tap to reveal answer
$\text{Each corresponding side ratio equals }k$. Each ratio $\frac{\text{side}_1}{\text{side}_2} = k$ for all pairs.
$\text{Each corresponding side ratio equals }k$. Each ratio $\frac{\text{side}_1}{\text{side}_2} = k$ for all pairs.
← Didn't Know|Knew It →
What does $\text{CPCTC}$ stand for in triangle congruence proofs?
What does $\text{CPCTC}$ stand for in triangle congruence proofs?
Tap to reveal answer
$\text{Corresponding parts of congruent triangles are congruent}$. Standard definition of the CPCTC theorem.
$\text{Corresponding parts of congruent triangles are congruent}$. Standard definition of the CPCTC theorem.
← Didn't Know|Knew It →
Identify the missing reason: If $\triangle ABC\cong\triangle DEF$, then $\angle B\cong\angle E$ by what principle?
Identify the missing reason: If $\triangle ABC\cong\triangle DEF$, then $\angle B\cong\angle E$ by what principle?
Tap to reveal answer
$\text{CPCTC}$. Corresponding Parts of Congruent Triangles are Congruent.
$\text{CPCTC}$. Corresponding Parts of Congruent Triangles are Congruent.
← Didn't Know|Knew It →
Find $x$ if $\triangle ABC\sim\triangle DEF$, $\frac{AB}{DE}=\frac{2}{5}$, and $DE=20$ with $AB=x$.
Find $x$ if $\triangle ABC\sim\triangle DEF$, $\frac{AB}{DE}=\frac{2}{5}$, and $DE=20$ with $AB=x$.
Tap to reveal answer
$x=\frac{2}{5}\cdot 20=8$. Given ratio tells us $AB = \frac{2}{5} \cdot DE$.
$x=\frac{2}{5}\cdot 20=8$. Given ratio tells us $AB = \frac{2}{5} \cdot DE$.
← Didn't Know|Knew It →
Find $x$ if $\triangle ABC\sim\triangle DEF$, $AB=9$, $DE=6$, and $AC=x$ corresponds to $DF=10$.
Find $x$ if $\triangle ABC\sim\triangle DEF$, $AB=9$, $DE=6$, and $AC=x$ corresponds to $DF=10$.
Tap to reveal answer
$x=10\cdot\frac{9}{6}=15$. Scale factor from ABC to DEF is $\frac{9}{6} = \frac{3}{2}$.
$x=10\cdot\frac{9}{6}=15$. Scale factor from ABC to DEF is $\frac{9}{6} = \frac{3}{2}$.
← Didn't Know|Knew It →
If $\triangle ABC\sim\triangle DEF$, which angle corresponds to $\angle C$?
If $\triangle ABC\sim\triangle DEF$, which angle corresponds to $\angle C$?
Tap to reveal answer
$\angle F$. Third angles in similarity statement correspond.
$\angle F$. Third angles in similarity statement correspond.
← Didn't Know|Knew It →
What is the similarity statement if $\angle A\cong\angle D$, $\angle B\cong\angle E$, and $\angle C\cong\angle F$?
What is the similarity statement if $\angle A\cong\angle D$, $\angle B\cong\angle E$, and $\angle C\cong\angle F$?
Tap to reveal answer
$\triangle ABC\sim\triangle DEF$. Corresponding vertices are written in matching order.
$\triangle ABC\sim\triangle DEF$. Corresponding vertices are written in matching order.
← Didn't Know|Knew It →