1. Find the length of a side of an equilateral triangle whose altitude measures the square root of 300 cm.
2. In right triangle ABC, C is the right angle. If AC is 1 cm more than BC and AB is 2 cm more than BC, find the lengths of the three sides.
3. The length of a rhombus is 8 and one of the angles measures 120, find the lengths of each of the diagonals.
2. In right triangle ABC, C is the right angle. If AC is 1 cm more than BC and AB is 2 cm more than BC, find the lengths of the three sides.
3. The length of a rhombus is 8 and one of the angles measures 120, find the lengths of each of the diagonals.
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1. If we say that eqilateral triangle has a side of length s, then how do we calculate its altitude. Take right angle triangle that has one side 's' as hypotenuse and altitude and "halfside" 's/2' as legs. Pythagoras theorem says: s^2 = (s/2)^2+a^2 (a is altitude).
Fill sqrt(300) for 'a':
s^2 = (s^2)/4 = 300
s^2=400
s=20
Length of side is 20cm.
2. You can try using pythagorean theorem where you say that BC is x and AB is x+2 and AC is x+1 and then you have x^2 + (x+1)^2 = (x+2)^2 and you solve it. But you can also guess that it should probably be whole numbers and look at some right angle triangles with whole sides (https://en.wikipedia.org/wiki/Special_ri… and the most common (3,4,5) fits.
3. If one angle measures 120° then if you draw one of the diagonals, you split the rhombus to two equilateral triangles with side 8. So one diagonal has length 8.
The other diagonal is twice the altitude in such triangle. We know (using same method as in #1) that altitude in equilateral triangle is sqrt(3)/2 times larger than side. So two altitudes are sqrt(3) times side that is 8*sqrt(3) is length of second diagonal.
Fill sqrt(300) for 'a':
s^2 = (s^2)/4 = 300
s^2=400
s=20
Length of side is 20cm.
2. You can try using pythagorean theorem where you say that BC is x and AB is x+2 and AC is x+1 and then you have x^2 + (x+1)^2 = (x+2)^2 and you solve it. But you can also guess that it should probably be whole numbers and look at some right angle triangles with whole sides (https://en.wikipedia.org/wiki/Special_ri… and the most common (3,4,5) fits.
3. If one angle measures 120° then if you draw one of the diagonals, you split the rhombus to two equilateral triangles with side 8. So one diagonal has length 8.
The other diagonal is twice the altitude in such triangle. We know (using same method as in #1) that altitude in equilateral triangle is sqrt(3)/2 times larger than side. So two altitudes are sqrt(3) times side that is 8*sqrt(3) is length of second diagonal.