Congruence and Similarity to Solve Problems - Geometry
Card 1 of 30
In right $\triangle ABC$ with $\angle C=90^\circ$, if $CD$ is the altitude to $AB$, what triangles are similar?
In right $\triangle ABC$ with $\angle C=90^\circ$, if $CD$ is the altitude to $AB$, what triangles are similar?
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$\triangle ABC\sim\triangle ACD\sim\triangle BCD$. Altitude to hypotenuse creates three similar triangles.
$\triangle ABC\sim\triangle ACD\sim\triangle BCD$. Altitude to hypotenuse creates three similar triangles.
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In right $\triangle ABC$, altitude $CD$ meets $AB$ at $D$. If $AD=9$ and $DB=16$, what is $CD$?
In right $\triangle ABC$, altitude $CD$ meets $AB$ at $D$. If $AD=9$ and $DB=16$, what is $CD$?
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$12$. $CD = \sqrt{AD \cdot DB} = \sqrt{9 \cdot 16} = 12$.
$12$. $CD = \sqrt{AD \cdot DB} = \sqrt{9 \cdot 16} = 12$.
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In right $\triangle ABC$, $AD=4$ and $DB=5$ on hypotenuse $AB$. What is $AB$?
In right $\triangle ABC$, $AD=4$ and $DB=5$ on hypotenuse $AB$. What is $AB$?
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$9$. $AD + DB = 4 + 5 = 9$.
$9$. $AD + DB = 4 + 5 = 9$.
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Identify the name of the segment from a vertex perpendicular to the opposite side of a triangle.
Identify the name of the segment from a vertex perpendicular to the opposite side of a triangle.
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Altitude. Perpendicular segment from vertex to opposite side.
Altitude. Perpendicular segment from vertex to opposite side.
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What is the name of the segment joining midpoints of two sides of a triangle?
What is the name of the segment joining midpoints of two sides of a triangle?
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Midsegment. Connects midpoints of two triangle sides.
Midsegment. Connects midpoints of two triangle sides.
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What does the midsegment theorem state about a midsegment and the third side of a triangle?
What does the midsegment theorem state about a midsegment and the third side of a triangle?
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It is parallel and has half the third side’s length. Midsegment is parallel and half the length of third side.
It is parallel and has half the third side’s length. Midsegment is parallel and half the length of third side.
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In $\triangle ABC$, if $DE\parallel BC$ with $D\in AB$ and $E\in AC$, what similarity must hold?
In $\triangle ABC$, if $DE\parallel BC$ with $D\in AB$ and $E\in AC$, what similarity must hold?
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$\triangle ADE\sim\triangle ABC$. Parallel line creates similar triangles with same vertex order.
$\triangle ADE\sim\triangle ABC$. Parallel line creates similar triangles with same vertex order.
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In right $\triangle ABC$, $AB=20$ and $DB=5$. Using similarity, what is $BC$?
In right $\triangle ABC$, $AB=20$ and $DB=5$. Using similarity, what is $BC$?
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$10$. $BC = \sqrt{AB \cdot DB} = \sqrt{20 \cdot 5} = 10$.
$10$. $BC = \sqrt{AB \cdot DB} = \sqrt{20 \cdot 5} = 10$.
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In $\triangle ABC$, with $DE\parallel BC$, what proportionality relates the divided sides $AB$ and $AC$?
In $\triangle ABC$, with $DE\parallel BC$, what proportionality relates the divided sides $AB$ and $AC$?
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$\frac{AD}{DB}=\frac{AE}{EC}$. Parallel line divides sides proportionally (segments).
$\frac{AD}{DB}=\frac{AE}{EC}$. Parallel line divides sides proportionally (segments).
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In $\triangle ABC$, with $DE\parallel BC$, what ratio equality compares the smaller to the whole triangle on side $AB$ and $AC$?
In $\triangle ABC$, with $DE\parallel BC$, what ratio equality compares the smaller to the whole triangle on side $AB$ and $AC$?
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$\frac{AD}{AB}=\frac{AE}{AC}$. Parallel line creates proportional ratios (part to whole).
$\frac{AD}{AB}=\frac{AE}{AC}$. Parallel line creates proportional ratios (part to whole).
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In $\triangle ABC$, with $DE\parallel BC$, what ratio equality compares the smaller to the whole triangle on side $AB$ and base $BC$?
In $\triangle ABC$, with $DE\parallel BC$, what ratio equality compares the smaller to the whole triangle on side $AB$ and base $BC$?
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$\frac{AD}{AB}=\frac{DE}{BC}$. Similar triangles have proportional corresponding sides.
$\frac{AD}{AB}=\frac{DE}{BC}$. Similar triangles have proportional corresponding sides.
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In $\triangle ABC$, with $DE\parallel BC$, what ratio equality compares the smaller to the whole triangle on side $AC$ and base $BC$?
In $\triangle ABC$, with $DE\parallel BC$, what ratio equality compares the smaller to the whole triangle on side $AC$ and base $BC$?
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$\frac{AE}{AC}=\frac{DE}{BC}$. Similar triangles have proportional corresponding sides.
$\frac{AE}{AC}=\frac{DE}{BC}$. Similar triangles have proportional corresponding sides.
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What is the key conclusion of the converse: if $\frac{AD}{DB}=\frac{AE}{EC}$, then what must be true?
What is the key conclusion of the converse: if $\frac{AD}{DB}=\frac{AE}{EC}$, then what must be true?
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$DE\parallel BC$. Converse: proportional division implies parallel line.
$DE\parallel BC$. Converse: proportional division implies parallel line.
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In right $\triangle ABC$, $AB=18$ and $AD=8$. Using $AC^2=AB\cdot AD$, what is $AC$?
In right $\triangle ABC$, $AB=18$ and $AD=8$. Using $AC^2=AB\cdot AD$, what is $AC$?
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$12$. $AC = \sqrt{18 \cdot 8} = \sqrt{144} = 12$.
$12$. $AC = \sqrt{18 \cdot 8} = \sqrt{144} = 12$.
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In right $\triangle ABC$, $AB=18$ and $DB=2$. Using $BC^2=AB\cdot DB$, what is $BC$?
In right $\triangle ABC$, $AB=18$ and $DB=2$. Using $BC^2=AB\cdot DB$, what is $BC$?
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$6$. $BC = \sqrt{18 \cdot 2} = \sqrt{36} = 6$.
$6$. $BC = \sqrt{18 \cdot 2} = \sqrt{36} = 6$.
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In right $\triangle ABC$, $AD=1$ and $DB=9$. Using $CD^2=AD\cdot DB$, what is $CD$?
In right $\triangle ABC$, $AD=1$ and $DB=9$. Using $CD^2=AD\cdot DB$, what is $CD$?
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$3$. $CD = \sqrt{1 \cdot 9} = \sqrt{9} = 3$.
$3$. $CD = \sqrt{1 \cdot 9} = \sqrt{9} = 3$.
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What theorem states that a line parallel to one side of a triangle divides the other two sides proportionally?
What theorem states that a line parallel to one side of a triangle divides the other two sides proportionally?
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Triangle proportionality theorem. States parallel line divides other sides proportionally.
Triangle proportionality theorem. States parallel line divides other sides proportionally.
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What is the converse theorem called: if a line divides two sides proportionally, then it is parallel to the third side?
What is the converse theorem called: if a line divides two sides proportionally, then it is parallel to the third side?
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Converse of the triangle proportionality theorem. If division is proportional, then line is parallel to third side.
Converse of the triangle proportionality theorem. If division is proportional, then line is parallel to third side.
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What is the standard similarity criterion used to prove the triangle proportionality theorem?
What is the standard similarity criterion used to prove the triangle proportionality theorem?
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AA similarity. Angle-Angle similarity proves triangles are similar.
AA similarity. Angle-Angle similarity proves triangles are similar.
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What similarity criterion is commonly used to prove the Pythagorean Theorem via an altitude in a right triangle?
What similarity criterion is commonly used to prove the Pythagorean Theorem via an altitude in a right triangle?
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AA similarity. AA similarity establishes similar triangles in altitude proof.
AA similarity. AA similarity establishes similar triangles in altitude proof.
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What is the Pythagorean Theorem statement for a right triangle with legs $a,b$ and hypotenuse $c$?
What is the Pythagorean Theorem statement for a right triangle with legs $a,b$ and hypotenuse $c$?
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$a^2+b^2=c^2$. Sum of squares of legs equals square of hypotenuse.
$a^2+b^2=c^2$. Sum of squares of legs equals square of hypotenuse.
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Find the missing value: if $AC^2=AB\cdot AD$, $AB=15$, and $AD=6$, what is $AC$?
Find the missing value: if $AC^2=AB\cdot AD$, $AB=15$, and $AD=6$, what is $AC$?
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$3\sqrt{10}$. $AC = \sqrt{15 \cdot 6} = \sqrt{90} = 3\sqrt{10}$.
$3\sqrt{10}$. $AC = \sqrt{15 \cdot 6} = \sqrt{90} = 3\sqrt{10}$.
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In a triangle, if a segment is parallel to one side, what angle relationship justifies AA similarity?
In a triangle, if a segment is parallel to one side, what angle relationship justifies AA similarity?
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Corresponding angles are congruent. Parallel lines create congruent corresponding angles.
Corresponding angles are congruent. Parallel lines create congruent corresponding angles.
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What is the similarity-based relationship between corresponding sides of similar triangles?
What is the similarity-based relationship between corresponding sides of similar triangles?
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Corresponding side lengths are proportional. Fundamental property of similar triangles.
Corresponding side lengths are proportional. Fundamental property of similar triangles.
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In a right triangle, what is the name of the side opposite the right angle?
In a right triangle, what is the name of the side opposite the right angle?
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Hypotenuse. Longest side in right triangle, opposite 90° angle.
Hypotenuse. Longest side in right triangle, opposite 90° angle.
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In right $\triangle ABC$ with $\angle C=90^\circ$, if $CD$ is the altitude to $AB$, what triangles are similar?
In right $\triangle ABC$ with $\angle C=90^\circ$, if $CD$ is the altitude to $AB$, what triangles are similar?
Tap to reveal answer
$\triangle ABC\sim\triangle ACD\sim\triangle BCD$. Altitude to hypotenuse creates three similar triangles.
$\triangle ABC\sim\triangle ACD\sim\triangle BCD$. Altitude to hypotenuse creates three similar triangles.
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In right $\triangle ABC$ with altitude $CD$ to hypotenuse $AB$, what is the geometric mean relation for $CD$?
In right $\triangle ABC$ with altitude $CD$ to hypotenuse $AB$, what is the geometric mean relation for $CD$?
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$CD^2=AD\cdot DB$. Altitude is geometric mean of hypotenuse segments.
$CD^2=AD\cdot DB$. Altitude is geometric mean of hypotenuse segments.
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In right $\triangle ABC$ with altitude $CD$ to $AB$, what relation links leg $AC$ to $AB$ and $AD$?
In right $\triangle ABC$ with altitude $CD$ to $AB$, what relation links leg $AC$ to $AB$ and $AD$?
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$AC^2=AB\cdot AD$. Leg squared equals hypotenuse times adjacent segment.
$AC^2=AB\cdot AD$. Leg squared equals hypotenuse times adjacent segment.
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In right $\triangle ABC$ with altitude $CD$ to $AB$, what relation links leg $BC$ to $AB$ and $DB$?
In right $\triangle ABC$ with altitude $CD$ to $AB$, what relation links leg $BC$ to $AB$ and $DB$?
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$BC^2=AB\cdot DB$. Leg squared equals hypotenuse times adjacent segment.
$BC^2=AB\cdot DB$. Leg squared equals hypotenuse times adjacent segment.
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Using $AC^2=AB\cdot AD$ and $BC^2=AB\cdot DB$, what identity follows after adding?
Using $AC^2=AB\cdot AD$ and $BC^2=AB\cdot DB$, what identity follows after adding?
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$AC^2+BC^2=AB\cdot(AD+DB)$. Adding the two leg-segment relations.
$AC^2+BC^2=AB\cdot(AD+DB)$. Adding the two leg-segment relations.
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