Practice Questions

In $\triangle PQR$, a line segment $ST$ is drawn parallel to $QR$ such that $S$ is on $PQ$ and $T$ is on $PR$. If $PS = 4$ cm, $SQ = 6$ cm, and $PT = 5$ cm, calculate the length of $TR$.

In $\triangle ABC$, $\angle B = 75^\circ$ and $\angle C = 50^\circ$. In $\triangle XYZ$, $\angle Y = 75^\circ$ and $\angle Z = 50^\circ$. Examine if $\triangle ABC \sim \triangle XYZ$ and state the criterion used.

Let $ABCD$ be a trapezium with $AB \| DC$. If the diagonals $AC$ and $BD$ intersect at point $O$, create a proof based on similarity to show that $\frac{AO}{OC} = \frac{BO}{OD}$.

Define similar figures.

Formulate a proof for the statement: If a line is drawn through the mid-point of one side of a triangle parallel to another side, it bisects the third side.

In a triangle, a line segment is drawn parallel to one side, dividing the other two sides. If the segments of one side are $x$ and $x-2$, and the corresponding segments of the other side are $x+2$ and $x-1$, calculate the value of $x$.

Are all equilateral triangles similar? Explain your answer in one sentence.

Propose a complete proof for the theorem stating that the ratio of the perimeters of two similar triangles is equal to the ratio of their corresponding sides.

State the Basic Proportionality Theorem, also known as Thales Theorem.

The sides of $\triangle LMN$ are 6 cm, 8 cm, and 10 cm. The sides of $\triangle DEF$ are 9 cm, 12 cm, and 15 cm. Analyze if the triangles are similar. If yes, write the similarity relation in correct correspondence.

In $\triangle XYZ$, point $A$ is on side $XY$ and point $B$ is on side $XZ$. If $XA = 3$ cm, $AY = 5$ cm, $XB = 4.5$ cm and $BZ = 7.5$ cm, determine if $AB \| YZ$.

In $\triangle PQR$, $PQ = 5$ cm, $PR = 10$ cm, and $\angle P = 50^\circ$. In $\triangle STU$, $ST = 7.5$ cm, $SU = 15$ cm, and $\angle S = 50^\circ$. Examine if the two triangles are similar.

The diagonals of a trapezium $PQRS$ with $PQ \| SR$ intersect at point $O$. If $PO = (3x - 1)$ cm, $OR = (5x - 3)$ cm, $QO = (2x + 1)$ cm, and $OS = (6x - 5)$ cm, solve for $x$.

A lamp post is 4.5 m high. A girl of height 1.5 m stands at a distance of 6 m from the base of the lamp post. Calculate the length of her shadow cast on the ground.

In the given figure, $LM \| CB$ and $LN \| CD$. If $AM = 4$ cm, $MB = 6$ cm, and $AN = 5$ cm, calculate the length of $AD$.

In a right triangle $ABC$, right-angled at $B$, a perpendicular $BD$ is drawn from $B$ to the hypotenuse $AC$. Justify that $BD^2 = AD \cdot DC$.

List the three criteria for the similarity of two triangles.

If two triangles $\triangle ABC \sim \triangle DEF$, what can you state about their corresponding angles and corresponding sides?

Explain the difference between congruent figures and similar figures. Provide one example of a pair of congruent figures and one example of a pair of similar figures.

Describe the SSS (Side-Side-Side) similarity criterion. If $\triangle LMN \sim \triangle XYZ$ by SSS similarity, write the relationship between their corresponding sides.

State the converse of the Basic Proportionality Theorem and draw a diagram to illustrate it.

Given two equilateral triangles, $\triangle ABC$ with a side length of 5 cm and $\triangle PQR$ with a side length of 10 cm. Explain why these two triangles are similar, referring to the definition of similar polygons.

State and explain the SAS (Side-Angle-Side) similarity criterion for two triangles. Draw two triangles, $\triangle ABC$ and $\triangle PQR$, that are similar by this criterion. Label the corresponding equal angles and the sides that are in proportion, and write the similarity in symbolic form.

Describe the concept of a scale factor in the context of similar polygons. If quadrilateral ABCD is similar to quadrilateral PQRS and the ratio of corresponding sides $\frac{AB}{PQ} = \frac{2}{3}$, what does this scale factor tell you about the relative sizes of the two figures? What would be the ratio of their perimeters?

Justify whether the Side-Side-Angle (SSA) condition is sufficient for the similarity of two triangles. If not, formulate a counterexample.

Critique the following statement: 'If the corresponding altitudes of two triangles are proportional, the triangles must be similar.' Justify your reasoning.

In $\triangle ABC$, $D$ is the mid-point of $BC$. The angle bisectors of $\angle ADB$ and $\angle ADC$ meet $AB$ and $AC$ at $E$ and $F$ respectively. Formulate a proof to show that $EF \| BC$.

Design a proof to show that the line segment joining the mid-points of any two sides of a triangle is parallel to the third side and is half of it.

In $\triangle ABC$, altitudes $AD$ and $CE$ intersect at point $P$. Formulate a proof to show that $PA \cdot PD = PC \cdot PE$.

Propose a formal proof for the theorem stating that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding altitudes.

Explain the AA (Angle-Angle) similarity criterion. Why is it sufficient to show that only two pairs of corresponding angles are equal for two triangles to be similar?

Summarize the two conditions required for any two polygons with the same number of sides to be similar. Provide one example of a pair of quadrilaterals where corresponding angles are equal but they are not similar, and another example where corresponding sides are in the same ratio but they are not similar.

Two poles of height $a$ meters and $b$ meters are $p$ meters apart. Prove that the height ($h$) of the intersection point of the lines joining the top of each pole to the foot of the opposite pole is given by $h = \frac{ab}{a+b}$ meters.

Identify if a square with a side length of 4 cm and a rectangle with a length of 5 cm and a breadth of 4 cm are similar. Explain your reasoning based on the conditions for similarity of polygons.

A circle is drawn with side $BC$ of $\triangle ABC$ as its diameter. The circle intersects sides $AB$ and $AC$ at points $D$ and $E$ respectively. Devise a proof to show that $\triangle ADE \sim \triangle ABC$.

In $\triangle PQR$, right-angled at $Q$, an altitude $QS$ is drawn to the hypotenuse $PR$. If $PS = 4$ cm and $SR = 9$ cm, calculate the length of the altitude $QS$.

In the figure, $\triangle ABE \cong \triangle ACD$. Analyze the relationship between $\triangle ADE$ and $\triangle ABC$ and demonstrate that they are similar.

Propose a method, based on triangle similarity, to determine the height of a tall object (like a flagpole) using only a mirror and a measuring tape.

Using the principles of similar triangles, create a proof for the Pythagoras Theorem.

In an equilateral triangle $ABC$, $D$ is a point on side $BC$ such that $BD = \frac{1}{3}BC$. Justify that $9AD^2 = 7AB^2$.

Explain the relationship between congruent figures and similar figures. Are all congruent figures similar? Are all similar figures congruent? Justify your answers with a detailed explanation and provide examples using circles and squares.