T2T4 Review — Deductive Geometry — Grade 10

Congruence tests. Fluency

(a) ▵ABC and ▵PQR: AB = PQ = 9, ∠A = ∠P = 55°, AC = PR = 7. Congruent? State test.
(b) ▵DEF and ▵XYZ: DE = XY = 6, EF = YZ = 8, FD = ZX = 10. Congruent? State test.
(c) ∠A = ∠D = 90°, hypotenuse AB = DE = 13, leg BC = EF = 5. Congruent?
(d) ∠P = ∠X = 40°, ∠Q = ∠Y = 80°, QR = YZ = 7 (non-corresponding sides — check carefully). Are these congruent?

Scale factors and similarity. Fluency

(a) Similar shapes with sides 8 and 12. Find the area ratio.
(b) Similar solids: volume ratio 64 : 27. Find the length ratio.
(c) ▵ABC ~ ▵PQR, AB = 15, PQ = 10, BC = 12. Find QR.
(d) A map scale is 1 : 25 000. Two points are 3.6 cm apart on the map. Find the real distance in km.

Angle facts and parallel lines. Fluency

(a) Co-interior angles: (3x + 12)° and (5x − 8)°. Find x if lines are parallel.
(b) Corresponding angles (2y + 15)° and (4y − 25)°. Find y.
(c) Exterior angle of a triangle = 132°, one interior non-adjacent angle = 79°. Find the other.
(d) ABCD is a parallelogram. ∠A = (2x + 10)°, ∠C = (3x − 20)°. Find x and ∠A.

Quadrilateral identification. Fluency

(a) A quadrilateral with all sides equal and one right angle. What is it?
(b) A parallelogram with perpendicular diagonals. What is it?
(c) A quadrilateral with exactly one pair of parallel sides and equal diagonals. What is it?
(d) A quadrilateral with diagonals that are perpendicular and one bisects the other (but not both bisected). What is it?

Similar triangles in a diagram. Understanding

Two triangles share a common vertex at A. ▵ADE has AD = 8, AE = 6, DE = 10. ▵ABC has AB = 12, AC = 9, BC = 15. Points D and E lie on AB and AC respectively.

(a) Find the ratios AD/AB, AE/AC, and DE/BC.
(b) State why ▵ADE ~ ▵ABC and the test used.
(c) Prove DE ∥ BC.
(d) Find the scale factor from ▵ABC to ▵ADE, and hence the ratio of their areas.

Proof involving a parallelogram. Understanding

PQRS is a parallelogram. The diagonal PR and the diagonal QS intersect at T.

(a) Prove ▵PQT ≅ ▵RST.
(b) Hence prove that the diagonals bisect each other (PT = RT and QT = ST).
(c) If PQ = 9 cm and ∠PQS = 35°, find ∠QSR.
(d) If ∠QPS = 62° and ∠PSR = 118°, confirm PQRS is a parallelogram.

Intercept theorem applications. Understanding

(a) In ▵ABC, D on AB with AD = 5, DB = 3. E on AC with AE = 10. Find EC if DE ∥ BC.
(b) Three parallel lines cut two transversals. Intercepts on first: 4, 6. First intercept on second: 5. Find the second intercept.
(c) A shelf of length 90 cm is divided by supports into sections proportional to 2 : 3 : 4. Find the length of each section.
(d) In a trapezium ABCD (AB ∥ DC), the diagonals meet at E. AB = 8, DC = 12. Find AE/EC.

Coordinate geometry proof. Understanding

Vertices A(1,2), B(5,2), C(7,6), D(3,6).

(a) Find the length of all four sides.
(b) Find the slope of each side. What type of quadrilateral is ABCD?
(c) Find the lengths of the diagonals AC and BD.
(d) Is ABCD a rhombus? Justify.

Shadow and height problem. Problem Solving

At the same time of day, a 1.8-m person casts a shadow of 2.4 m. A building casts a shadow of 32 m. Both shadows are measured on level ground with the sun at the same angle.

(a) Set up the similar triangle ratio.
(b) Find the height of the building.
(c) At a different time of day, the person’s shadow is 4.2 m. Find the new shadow length of the building.
(d) A tree 15 m tall casts a shadow of length s at the same time as (b). Find s.

Trapezium proof and measurement. Problem Solving

ABCD is a trapezium with AB ∥ DC, AB = 10 cm, DC = 6 cm, height = 8 cm. M is the midpoint of AD and N is the midpoint of BC.

(a) MN is the midsegment of the trapezium. State its length (midsegment = average of parallel sides).
(b) Prove that MN ∥ AB by showing M and N have the same y-coordinate in a suitable coordinate system.
(c) Find the area of the trapezium.
(d) The trapezium is split into two triangles by diagonal AC. The triangles share the same height. Show that their areas are in ratio AB : DC = 10 : 6 = 5 : 3.

Scale model application. Problem Solving

An architect builds a model of a building at 1 : 200 scale. The real building is 48 m tall and has a floor area of 2400 m².

(a) Find the height of the model in cm.
(b) Find the floor area of the model in cm².
(c) The real building has a volume of 120 000 m³. Find the volume of the model in cm³.
(d) If the model is made from polystyrene of density 0.05 g/cm³, find the mass of the model in kg.

Optimisation using similar triangles. Problem Solving

A ladder of length 5 m leans against a wall. The base is x metres from the wall. The ladder forms a right-angled triangle with the wall and ground.

(a) Write the height h of the ladder up the wall in terms of x.
(b) A small similar triangle is formed by a horizontal shelf 1 m from the wall and the section of wall below it. If the shelf is at height y, write the similar triangle ratio in terms of x, y, and h.
(c) Find h and y when x = 3.
(d) If x = 4, find h and y. Which position (x=3 or x=4) gives the greater shelf height y?

Topic Review — Deductive Geometry