Triangles

Are all circles similar?

If a line divides any two sides of a triangle in the same ratio, then the line must be ____ to the third side.

In the diagram, $\triangle ABC \cong \triangle DEF$. Which side in $\triangle DEF$ corresponds to side $BC$?

Given two squares, one with side $4$ cm and another with side $7$ cm. These two squares are:

In $\triangle ABC$, $D$ and $E$ are points on sides $AB$ and $AC$ such that $DE \parallel BC$. If $AD/DB = 3/4$ and $AC = 14$ cm, find $AE$.

Two triangles are similar by SAS similarity if two sides of one triangle are proportional to two sides of another triangle and...

A vertical pole of length $6$ m casts a shadow $4$ m long on the ground and at the same time a tower casts a shadow $28$ m long. Find the height of the tower.

Given $\triangle ABC \sim \triangle PQR$ with $\frac{Area(\triangle ABC)}{Area(\triangle PQR)} = \frac{9}{25}$. If $BC = 12$ cm, what is the length of $QR$?

A shadow cast by a $2$ m tall pole is $3$ m long. At the same time, a tree casts a shadow $15$ m long. If the triangles formed by the pole/shadow and tree/shadow are similar, find the height of the tree.

In a geometry project, a student uses a scale factor of $k = 1$ to map $\triangle T1$ to $\triangle T2$. If all corresponding angles are equal, what can be concluded?

In $\triangle ABC$, $DE \parallel BC$. If $AD = 2.4$ cm, $DB = 3.6$ cm and $AC = 5$ cm, find the length of $EC$.

In the given figure, $ABCD$ is a quadrilateral. If $P, Q, R, S$ are points on $AB, BC, CD, DA$ respectively such that $PQRS$ is a parallelogram, which theorem helps in proving this using the diagonals of $ABCD$?

In $\triangle ABC$, $D$ and $E$ are points on $AB$ and $AC$ respectively. If $AB = 12$ cm, $AD = 4$ cm, $AC = 18$ cm and $AE = 6$ cm, show the relationship between $DE$ and $BC$.

In $\triangle ABC$, $DE \parallel BC$ and $AD/DB = 5/3$. If $AC = 5.6$ cm, find the length of $AE$.

In the given figure, $PQ \parallel RS$ and $PR \parallel QS$. If $T$ is a point on $PR$ such that a line through $T$ parallel to $RS$ meets $PS$ at $U$, and $PU = 4\text{ cm}$, $US = 6\text{ cm}$, and $PT = 3\text{ cm}$, find the length of $TR$ using the Basic Proportionality Theorem in $\triangle PSR$.

In $\triangle ABC$ shown below, $DE \parallel BC$. If $AD = x$, $DB = x - 2$, $AE = x + 2$, and $EC = x - 1$, find the value of $x$.

In $\triangle ABC$, $D$ and $E$ are points on sides $AB$ and $AC$ such that $DE \parallel BC$. If $AD = x$, $DB = x+1$, $AE = x+3$, and $EC = x+5$, find the value of $x$.

In a geometry problem, $D$ and $E$ are points on sides $AB$ and $AC$ of $\triangle ABC$ such that $DE \parallel BC$. If $AD = 2$ cm, $DB = 3$ cm, and the area of $\triangle ADE$ is $8$ $\text{cm}^2$, what is the area of the quadrilateral $DBCE$?

A metal frame is in the shape of $\triangle ABC$. A horizontal support bar $PQ$ is welded such that $P$ is on $AB$, $Q$ is on $AC$, and $PQ \parallel BC$. If $AP = 4$ cm, $PB = 6$ cm, and $BC = 15$ cm, calculate the length of the support bar $PQ$.

Given a triangle $ABC$ where $D$ is a point on $AB$ and $E$ is a point on $AC$ such that $DE \parallel BC$. If $AD/AB = 1/4$ and $AE = 2$ cm, what is the length of $CE$?

In $\triangle ABC$, $P$ and $Q$ are points on sides $AB$ and $AC$ such that $AP=5$, $PB=10$, $AQ=3$, $QC=6$. Which statement is true by the Converse of Thales Theorem?

If in $\triangle ABC$, $AD/DB = 1/2$ and $AE = 2$, $EC = 4$, then segment $DE$ must be:

In $\triangle ABC$, $D$ and $E$ are on $AB$ and $AC$. If $AB = 10$, $AD = 4$, $AC = 15$, and $AE = 6$, what is the relationship between $DE$ and $BC$?

In $\triangle XYZ$, $M$ and $N$ are points on $XY$ and $XZ$. If $XM = 4$, $MY = 6$, $XN = 6$, and $NZ = 9$, then which ratio confirms $MN \parallel YZ$?

In $\triangle ABC$, $D$ and $E$ are points on sides $AB$ and $AC$ respectively. If $AB = 7.2$, $AD = 1.8$, $AC = 6.4$, and $AE = 1.6$, then $DE$ is:

Two triangles $ABC$ and $DBC$ are on the same base $BC$. If a line parallel to $BC$ intersects $AB, AC, DB, DC$ at $P, Q, R, S$ respectively, then $PQ/RS$ is:

In $\triangle ABC$, $DE \parallel BC$ is to be established. We are given $AD = x + 3$, $DB = 3x + 19$, $AE = x$, and $EC = 3x + 4$. Find $x$.

In a trapezoid $ABCD$ where $AB \parallel CD$, diagonals $AC$ and $BD$ intersect at $O$. If a line through $O$ intersects $AD$ at $P$ and $BC$ at $Q$ such that $AP/PD = BQ/QC$, what can be said about $PQ$?

In $\triangle DEF$, $M$ is a point on $DE$ and $N$ is a point on $DF$. If $DM = 4$ cm, $ME = 6$ cm, $DN = 5$ cm, and $NF = 7.5$ cm, which of the following is true?

An architect is designing a triangular window. On the left frame ($AB$), a mark $D$ is placed $60$ cm from the top vertex $A$. The total length $AB$ is $100$ cm. On the right frame ($AC$), a mark $E$ is placed $45$ cm from $A$. The total length $AC$ is $75$ cm. A horizontal bar is placed between $D$ and $E$. Is the bar parallel to the bottom frame $BC$?

If $\triangle ABC \sim \triangle DEF$ and $BC = 3$ cm, $EF = 4$ cm, $ar(\triangle ABC) = 54 \text{ cm}^2$, find $ar(\triangle DEF)$.

Two isosceles triangles have equal vertical angles and their areas are in the ratio $16:25$. Find the ratio of their corresponding heights.

If the corresponding sides of two similar triangles are in the ratio $2:3$, then the ratio of their corresponding altitudes is:

Which of the following sets of side lengths represents two similar triangles?

In the figure, $AB \parallel CD \parallel EF$. If $AB = 6$ cm, $EF = 10$ cm, and $BD = 4$ cm, find $CD$ (approximately).

In the figure, $DE \parallel AC$ and $DF \parallel AE$. Which of the following relations is correct?

In $\triangle ABC$, $\angle B = 90^circ$ and $BD \perp AC$. Which of the following is NOT true?

In the triangle $ABC$ shown, $D$ is a point on $BC$ such that $\angle ADC = \angle BAC$. If $CA = 12$ cm and $CB = 16$ cm, find the length of $CD$ using the AA similarity criterion.

A girl of height $90$ cm is walking away from the base of a lamp-post at a speed of $1.2$ m/s. If the lamp is $3.6$ m above the ground, find the length of her shadow after $4$ seconds.

In a $\triangle ABC$, points $P$ and $Q$ are on $AB$ and $AC$ respectively such that $AP = 2$ cm, $PB = 4$ cm, $AQ = 3$ cm and $QC = 6$ cm. If $BC = 12$ cm, find $PQ$ using the SAS similarity criterion.

In the given figure, $\triangle ABC$ is a triangle where $DE \parallel BC$. If $AD = 3$ cm, $DB = 5$ cm, and $AE = 4.5$ cm, find the length of $EC$.

In $\triangle ABC$, $\angle B = 90^{\circ}$, $AB = 5$ cm and $BC = 12$ cm. Find the length of $AC$.

Two triangles are congruent if they have the same ________.

In the figure, $XY || BC$. If $AX = 1$ cm, $XB = 3$ cm, and $BC = 6$ cm, find $XY$.

In the figure shown, $\triangle PQR$ is similar to $\triangle XYZ$ ($\triangle PQR \sim \triangle XYZ$). If the ratio of their corresponding sides is $\frac{PQ}{XY} = \frac{2}{3}$ and the area of $\triangle PQR$ is $32$ cm$^2$, determine the area of $\triangle XYZ$.

In the given figure, $P$ and $Q$ are points on sides $AB$ and $AC$ such that $PQ || BC$. If $AP = 2.4$ cm, $AQ = 2$ cm, $QC = 3$ cm, find $AB$.

The areas of two similar triangles are $81 \text{ cm}^2$ and $49 \text{ cm}^2$ respectively. If the altitude of the bigger triangle is $4.5$ cm, find the altitude of the smaller triangle.

In $\triangle ABC$, $DE \parallel BC$. If $AD = x$, $DB = x-2$, $AE = x+2$, and $EC = x-1$, find the value of $x$.

$D$ and $E$ are points on the sides $CA$ and $CB$ respectively of a triangle $ABC$ right angled at $C$. Which of the following identity is correct regarding the lengths of segments $AE, BD, AB$ and $DE$?

In a trapezium $ABCD$ with $AB \parallel DC$, the diagonals $AC$ and $BD$ intersect at $O$. If $AB = 3 DC$, find the ratio of the areas of $\triangle AOB$ and $\triangle COD$.

In the given figure, $PQ \parallel BC$. If $AP = 2.4$ cm, $AQ = 2$ cm, $QC = 3$ cm, find $PB$.

In $\triangle ABC$, $D$ and $E$ are points on $AB$ and $AC$ such that $DE \parallel BC$ and $AD = 1/3 AB$. If $BC = 15$ cm, then $DE$ is:

All squares are:

In the figure, if $\triangle ABC \sim \triangle DEF$, then the value of $x$ is:

In $\triangle ABC$, $DE \parallel BC$ with $D$ on $AB$ and $E$ on $AC$. If $AD = x$, $DB = x - 2$, $AE = x + 2$, and $EC = x - 1$, find the value of $x$.

In the figure, $O$ is the point of intersection of two chords $AB$ and $CD$ such that $\triangle AOC \sim \triangle DOB$. If $AC = 4$ cm, $BD = 6$ cm, $OC = 3$ cm, find $OB$.

If $\triangle ABC \sim \triangle QRP$, $area(\triangle ABC)/area(\triangle PQR) = 9/4$, $AB = 18$ cm and $BC = 15$ cm, then $PR$ is:

An isosceles triangle $ABC$ with $AB=AC$ is similar to $\triangle PQR$. If $AB=5$ cm and $BC=8$ cm, and the corresponding side $PQ = 7.5$ cm, find the length of $QR$.

In the figure, $DE \parallel BC$. If $AD : DB = 3 : 5$ and $AC = 5.6$ cm, find the length of $AE$.

Given $\triangle ABC \sim \triangle DEF$. If the perimeter of $\triangle ABC$ is $30$ cm and the perimeter of $\triangle DEF$ is $18$ cm, and $BC = 9$ cm, then the length of $EF$ is:

If $\triangle ABC \sim \triangle DEF$, $BC = 3$ cm, $EF = 4$ cm, and area of $\triangle ABC = 54$ cm$^2$, find the area of $\triangle DEF$.

In $\triangle ABC$, $P$ and $Q$ are points on sides $AB$ and $AC$ respectively such that $PQ \parallel BC$. If $AP = 2.4$ cm, $AB = 6$ cm, and $AC = 5$ cm, find $AQ$.

Which criterion of similarity is used if all corresponding angles of two triangles are equal?

In $\triangle ABC$, $DE \parallel BC$. If $AD = x$, $DB = x-2$, $AE = x+2$, and $EC = x-1$, find the value of $x$.

In the given figure, $DE \parallel BC$. If $AD = 2.4\text{ cm}$, $DB = 3.6\text{ cm}$, and $AC = 5\text{ cm}$, find the length of $AE$.

In $\triangle ABC$, $AD$ is the perpendicular bisector of $BC$. If $\triangle ABC \sim \triangle DEF$, and $AD = 8$ cm, $BC = 12$ cm, $EF = 3$ cm, find the altitude of $\triangle DEF$.

Two poles of height $7$ m and $12$ m stand vertically on a plane ground. If the distance between their feet is $12$ m, find the distance between their tops.

In $\triangle ABC$, $\angle B = 90^\circ$ and $BD \perp AC$. If $AC = 16$ cm and $AD = 4$ cm, find the length of $BD$ in cm.