Geometry and measures, Year 9: Pythagoras and angle reasoning

The hypotenuse

提示It is the only side that does not touch the square corner.

为什么Pythagoras' theorem is built around this side: its square equals the squares on the other two sides added together. Find it before doing any arithmetic, because it decides whether you add or subtract.

13

提示The diagonal cuts the rectangle into two triangles, each with a square corner.

为什么The width and length meet at a right angle, so the diagonal is the hypotenuse of a triangle with shorter sides 5 and 12. Then 5² + 12² = 25 + 144 = 169, and √169 = 13.

It must be right-angled

提示Think about the one special corner the theorem depends on.

为什么The rule only works when the two shorter sides meet at 90°, with c opposite that corner. If the angle is smaller than 90°, a² + b² comes out bigger than c²; if it is larger, a² + b² comes out smaller.

No: 5² + 6² = 61, but 8² = 64

提示Square all three lengths, then see whether the two smaller squares make the largest.

为什么In a right-angled triangle the two smaller squares add up to the largest exactly. Here 25 + 36 = 61, which misses 64, so no angle of this triangle is 90°.

225; 15 cm

提示The longest side's square is the total, so the missing square is what remains once the known one is removed.

为什么The theorem says 8² + (missing side)² = 17², and 17² = 289 while 8² = 64, so the missing square is 289 − 64. Taking the square root at the end turns that square back into a length.

8 m

提示The ladder itself faces the square corner where the wall meets the ground.

为什么The wall and the ground meet at a right angle, so the ladder is the hypotenuse. The height is a shorter side: 10² − 6² = 100 − 36 = 64, and √64 = 8.

Add them

提示Decide first whether the unknown side is the one facing the square corner.

为什么The sides that meet at the right angle are the two shorter sides, so the unknown is the hypotenuse: 81 + 144 = 225, giving 15 cm. You subtract only when the hypotenuse is one of the lengths you already know.

No side can be longer than the hypotenuse

提示Compare the size of the answer against the 15 cm already given.

为什么The hypotenuse faces the largest angle, so it is always the longest of the three sides. An answer bigger than it usually means the squares were added when they should have been subtracted.

3

提示The line splits the shape into two matching right-angled triangles — work out the short base of one of them first.

为什么Each half is a right-angled triangle with hypotenuse 5 and base 4, half of 8. The height is then √(5² − 4²) = √(25 − 16) = √9 = 3. Many problems hide a right-angled triangle like this inside another shape.

Alternate angles are equal

提示Picture the letter Z drawn across the two lines.

为什么A geometric argument gives the name of the fact used, not a sketch of a letter. "Alternate angles are equal", "corresponding angles are equal" and "co-interior angles add up to 180°" are the three parallel-line reasons.

The reason: angles in a triangle add up to 180°

提示The arithmetic is fine. Ask what would tell a reader where the starting number came from.

为什么Every claim in a geometric argument needs the fact that justifies it written beside it. Without it the reader cannot tell whether 180 came from a triangle, a straight line or a guess.

180

提示They sit inside a pair of parallel sides, on one side of the line joining them — the C shape.

为什么Each pair of neighbouring corners lies between two parallel sides, so the reason to write is "co-interior angles add up to 180°". Knowing one angle of a parallelogram therefore gives you all four.

70°; 110°

提示Share what is left of the triangle's total equally between B and C, then use the straight line through C.

为什么Base angles of an isosceles triangle are equal and the three angles total 180°, so each base angle is (180 − 40) ÷ 2. Angle ACD sits beside angle ACB on a straight line, so the two add up to 180°.

80

提示Three facts in a row: the straight line at B, the two equal sides, then the total for the triangle.

为什么Angle DBC = 180 − 130 = 50° (angles on a straight line). BD = CD, so the angles opposite those sides match and angle DCB = 50° (base angles of an isosceles triangle). Then angle BDC = 180 − 50 − 50 = 80° (angles in a triangle).