6.5 Secondary Research and you may Inequalities in a single Triangle

## six.step 3 Medians and you can Altitudes out of Triangles

Tell whether or not the orthocenter of your triangle towards given vertices are to the, with the, or outside the triangle. Following find the coordinates of your own orthocenter.

Explanation: The slope of the line HJ = $$\frac < 1> < 3>$$ = $$\frac < 5> < 2>$$ The slope of the perpendicular line = $$\frac < -2> < 5>$$ The perpendicular line is (y – 6) = $$\frac < -2> < 5>$$(x – 1) 5(y – 6) = -2(x – 1) 5y – 30 = -2x + 2 2x + 5y – 32 = 0 – (i) The slope of GJ = $$\frac < 1> < 3>$$ = $$\frac < -5> < 2>$$ The slope of the perpendicular line = $$\frac < 2> < 5>$$ The equation of perpendicular line (y – 6) = $$\frac < 2> < 5>$$(x – 5) 5(y – 6) = 2(x – 5) 5y – 30 = 2x – 10 2x – 5y + 20 = 0 – (ii) Equate both equations 2x + 5y – 32 = 2x – 5y + 20 10y = 52 y = 5.2 Substitute y = 5.2 in (i) 2x + 5(5.2) – 32 = 0 2x + 26 – 32 = 0 2x = 6 x = 3 The orthocenter is (3, 5.2) The orthocenter lies inside the triangle.

Explanation: The slope of LM = $$\frac < 5> < 0>$$ = $$\frac < 1> < 3>$$ The slope of the perpendicular line = -3 The perpendicular line is (y – 5) = -3(x + 8) y – 5 = -3x – 24 3x + y + 19 = 0 — (ii) The slope of KL = $$\frac < 3> < -6>$$ = -1 The slope of the perpendicular line = $$\frac < 1> < 2>$$ The equation of perpendicular line (y – 5) = $$\frac < 1> < 2>$$(x – 0) 2y – 10 = x — (ii) Substitute (ii) in (i) 3(2y – 10) + y + 19 = 0 6y – 30 + y + 19 = 0 7y – 11 = 0 y = $$\frac < 11> < 7>$$ x = -6 The othrocenter is (-6, -1) The orthocenter lies outside of the triangle

## 6.4 The latest Triangle Midsegment Theorem

Answer: Brand new midsegment out-of Abdominal = (-six, 6) This new midsegment away from BC = (-step 3, 4) This new midsegment from Ac = (-step 3, 6)

Explanation: The midsegment of AB = ($$\frac < -6> < 2>$$, antichat nedir $$\frac < 8> < 2>$$) = (-6, 6) The midsegment of BC = ($$\frac < -6> < 2>$$, $$\frac < 4> < 2>$$) = (-3, 4) The midsegment of AC = ($$\frac < -6> < 2>$$, $$\frac < 8> < 2>$$) = (-3, 6)

Answer: The fresh new midsegment of De- = (0, 3) The newest midsegment from EF = (2, 0) The fresh new midsegment regarding DF = (-step 1, -2)

Explanation: The midsegment of DE = ($$\frac < -3> < 2>$$, $$\frac < 1> < 2>$$) = (0, 3) The midsegment of EF = ($$\frac < 3> < 2>$$, $$\frac < 5> < 2>$$) = (2, 0) The midsegment of DF = ($$\frac < -3> < 2>$$, $$\frac < 1> < 2>$$) = (-1, -2)

Explanation: 4 + 8 > x 12 > x 4 + x > 8 x > 4 8 + x > 4 x > -4 4 < x < 12

Explanation: 6 + 9 > x 15 > x 6 + x > 9 x > 3 9 + x > 6 x > -3 3 < x < 15

Explanation: 11 + 18 > x 29 > x 11 + x > 18 x > 7 18 + x > 11 x > -7 7 < x < 29