070i - Finding a Shorter Side

Lesson 70 — Finding a Shorter Side; Guided Practice

Strand: Measurement | Descriptor: AC9M8M06 | Duration: 45 minutes

Learning Intentions

• To rearrange Pythagoras’ theorem to find the length of a leg given the hypotenuse and the other leg.
• To apply this to practical problems, expressing answers as surds or rounded decimals as appropriate.

Success Criteria

I can:

1. Rearrange to to solve for an unknown leg.
2. Correctly identify which given side is the hypotenuse before substituting.
3. Simplify or round answers appropriately, with correct units.
4. Solve applied problems (e.g. a ladder’s distance from a wall, the height of an isosceles triangle) by finding a shorter side.

Warmup

(5 minutes — mini whiteboards, rapid-fire)

1. In each of three triangle diagrams (some tilted), identify which given side is the hypotenuse.
2. Solve for : .
3. Solve for : (exact surd).
4. Why must you always subtract, never add, when the missing side is a leg?

Answers: 1. Always the side opposite the right angle, regardless of orientation; 2. ; 3. ; 4. Because the hypotenuse squared already includes both legs’ contributions — isolating one leg means removing the other leg’s share, which is subtraction.

Activities

Activity 1 — Guided Practice: Rearranging to Find a Leg (14 min)

I do:

I do (non-perfect-square case):

We do: Hypotenuse , one leg . Find the other leg.

You do:

1. Hypotenuse , leg .
2. Hypotenuse , leg (exact surd, then d.p.).
3. Hypotenuse , leg .
4. Hypotenuse , leg (rounded to d.p.).

Activity 2 — Applied Problems (13 min)

Pairs. Every final answer must carry correct units.

I do: A m ladder leans against a wall, reaching m up. How far is the foot of the ladder from the wall?

Problem 1. A rectangular gate has diagonal m and height m. Find its width.

Problem 2. An isosceles triangle has two equal sides of cm and a base of cm. Find its height, by splitting the base in half. (This uses the same “diagonal splits into congruent triangles” idea from Lessons 61 and 63 — here the altitude splits the isosceles triangle into two congruent right triangles.)

Problem 3. A rope is stretched from the top of a m flagpole to a point on the ground. The rope is m long. How far from the base of the pole does the rope meet the ground?

Activity 3 — Discrimination Task and Inquiry (8 min)

Pairs. Sort a mixed set of six problems into “find the hypotenuse” or “find a leg” before solving any of them — the goal is fast, reliable identification of the unknown’s role.

Then, as an applied inquiry task, tackle the isosceles-triangle-height problem below with full scaffolding:

A kite-shaped frame has two pairs of equal sides: cm and cm. Its longer diagonal (axis of symmetry) is cm. Find the length of the shorter diagonal.

Socratic scaffolding:

Prompt	Purpose
Understand: what shape and structure is this?	A kite — its axis of symmetry splits it into two pairs of congruent right triangles (recall Lesson 64’s kite symmetry result), and the diagonals meet at right angles.
What are the two right triangles’ hypotenuses?	The two different side lengths, cm and cm — one pair on each half of the kite.
Devise a plan	Find how the cm axis splits between the two pairs of triangles, then find the (shared) half of the short diagonal.
Carry out the plan	This needs more than one Pythagoras step — instead find each half of the short diagonal directly: half and half, where is the unknown split of the axis. Simplify the problem for this level by giving the axis split directly: cm and cm.
Carry out (simplified)	and .
Looking back	Both halves belong to the same short diagonal and, as expected from the kite’s symmetry, they match: each half is cm, so the shorter diagonal is cm.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Find the missing leg: hypotenuse , one leg .
2. Find the missing leg: hypotenuse , one leg , as an exact surd.
3. Round your answer to Q2 to decimal places.
4. A m ladder reaches m up a wall. How far is its foot from the wall?
5. Reasoning. Explain why you must always identify the hypotenuse correctly before deciding whether to add or subtract.

Answers: 1. ; 2. ; 3. ; 4. m; 5. If the hypotenuse is misidentified, the equation is set up backwards — subtracting when you should add (or vice versa) produces an impossible (negative) result or a wrong answer, since the hypotenuse’s square must always be the larger, isolated term.

Common Misconceptions

Misconception	How to pre-empt it
Adding instead of subtracting when solving for a leg.	Always write the rearranged equation () explicitly before substituting numbers.
Subtracting the wrong way round, e.g. .	Insist the larger value (hypotenuse squared) is always written first in the subtraction.
Misidentifying the hypotenuse in a tilted or unfamiliar diagram.	Always locate the right-angle marker first, then the side opposite it, regardless of orientation.
Getting a negative number under the square root and not noticing.	Treat a negative result as an automatic error flag — revisit which side was the hypotenuse.
Forgetting units in the final answer, or giving units for the wrong quantity.	Require every final answer to state both a number and a unit matching the original problem.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). A ladder m long rests against a wall, reaching m up. If the foot of the ladder is pulled out a further m, how far up the wall does the ladder now reach?

Answer

Original foot distance: m. New foot distance: m. New height: m. (A classic “swap” — the numbers exchange roles.)

E2 (Kangaroo style). An isosceles triangle has equal sides of cm and a base of cm. Find its area.

Answer

Height cm. Area .

E3 (Challenge). A right-angled triangle has hypotenuse and one leg equal to . Find the other leg in terms of , and identify the special triangle this describes.

Answer

Sides describe a –– triangle.

Homework

1. Find the missing leg: (a) hypotenuse , leg (b) hypotenuse , leg (c) hypotenuse , leg (exact surd) (d) hypotenuse , leg (rounded to d.p.).
2. A m ladder’s foot is placed so the ladder reaches m up a wall. Find the distance from the wall to the ladder’s foot.
3. An isosceles triangle has equal sides cm and base cm. Find its height.
4. A rectangle has diagonal cm and one side cm. Find the other side.
5. Reasoning. A classmate finds a leg by calculating instead of . Explain what mistake they’ve made and how you would help them check their own work.
6. Challenge. A ladder of length metres is placed so its foot is from the wall. Find, in terms of , an exact surd expression for how far up the wall it reaches, then simplify fully.

Answers: 1. (a) (b) (c) (d) ; 2. m; 3. cm; 4. cm; 5. They used the hypotenuse formula (adding) instead of the leg formula (subtracting) — a quick check is to confirm the answer is shorter than the hypotenuse, since a leg can never be the longest side; 6. .