This Pythagoras theorem problem?

There are more than one triangles here.

One triangle is on the ground. One side is "moving 1000 meters east", thus a line from west to east. The temple is then exactly in the north, so the second side of this triangle is a line south to north and the triangle has a right angle. The hypotenuse is a line pointing to the north-east, thus a 45 degree corner, thus the two small sides are equal in lenght.

There is another triangle: from the origin, the tempel's elevation is 45 degrees. This is another right angle triangle with two equal sides and a larger one.

45-45-90 triangles have sides 1:1:root(2)

The hypotenuse of the first triangle is thus 1000 meters times root(2) approx. 1414.21 meters.

This side is also one of the small sides of the second triangle, thus the second triangle's hypotenuse is root(2) times root(2) times 1000 meter, equalling 2000 meter.

You could also use Pythagoras.

hypotenuse triangle #1 is root( square(1000) + square(1000))

let 'a' is the answer of the previous computation, then the hypotenuse of triangle #2 is root( square(a) + square(a) )

equals 2000.

Edit: you say you don't understand the steps.

Think three dimensionally. You could do this in your living room.

First start drawing on the floor. From where you stand, draw a line a meter to your right. Then create a right angle, and go a meter forward. Go back where you started, you now have a triangle.

The sides of this triangle: 1, 1, and root(2).

You are back in your origin. Looking over the line with length root(2), you see the temple __above__ you. You have to look in a 45 degree angle. So, you have to create another triangle, perpendicular to the first one, going up towards the ceiling.

The sides of this new triangle are also in a ratio of 1:1:root(2). One of the two short sides of this triangle has size root(2). The other short side also has size root(2). The long side has length: root(2) times root(2), equals 2.

In case you don't see it yet: both triangles have the same corners, but the one flat on the floor is smaller than the one pointing to the ceiling.

In stead of using meters, the question was in kilometers. So, the answer of 2000 meter is the shortest line (a straight line) from the origin to the temple.

Edit #2 BTW where's my thumb-up ?!? :)

You ask why the hypotenuse of the first triangle has size 1000 times root(2), or why the triangle-on-paper has a 45 degree corner (really the same question).

Draw an X-axis and a Y-axis. The Y-axis is North (up) South (down); the X-axis is East (right) West (left).

Point O is the origin, where {x,y} = {0,0}

Direction north-east means: for every unit to the North, move an equal unit to the East.

You know you move 1000 meters to the east (let's call this point E). From O to E is a straight line, along the X-axis in a drawing, and its length is 1000 meter.

From point E, you can see the temple in the north. The temple's coordinates are thus {x,y} = {1000,something}.

Using "north-east of O" together with {1000,something} means the temple-on-a-hill has to be at the remaining point of the triangle-on-the-paper, at coordinates {1000,1000}. Say this is point H.

Side O-E is along the X-axis, side E-H is parallel to the Y-axis, thus the corner at E is 90 degrees.

A triangle with equal sides and a right angle is a 45-45-90 triangle, and its sides are in a 1:1:root(2) ratio. This means side O-H has to be 1000 times root(2). You can also use pythagoras to prove this; side E-H has to be of the same length as side O-E. Both are 1000 meter. side O-H is root(1000^2 + 1000^2) = 1414.21 (rounded). 1000 * root(2) is an exact answer, thus better than 1414.21

Now the second triangle. You need an extra axis, the Z-axis. The temple is on top of the hill. Let this be point T. you already know the bottom of the hill, H, and the origin, O.

The corner T-O-H is 45 degrees (given). Side T-H is parallel to the Z-axis, so corner T-H-O is 90 degrees. Another 45-45-90 triangle, thus a 1:1:root(2) ratio again.

O-H is 1000*root(2)

H-T has to be 1000*root(2) as well

Apply pythagoras, and get:

a^2 = (1000*root(2))^2 = 1000000*2

b^2 = (1000*root(2))^2 = 1000000*2

c^2 = a^2 + b^2 = 4000000

c = root(4000000) = 2000

No, it has to be solved with trigonometry because you know only one distance, not 2. When you have one distance and one angle,you trigonometry to find all the other distances.

Cosine = Adjacent / hypotenuse

in this case, Cos 45 = 1000 / Hypotenuse

Hyp = 1000 / Cos 45 = 1000 / 0.707 = 1414 meters.

But, in this case, since we have a 45 degree angle, that means that 2 sides are equal. Which means that we know 2 sides, so you can use the Pythagoras theorem

No. You need to use trigonometry.

Side A: One side of the triangle goes goes along the ground from initial point O to current point 0 with a length of 1000 meters. Side B: Goes from current point O to the temple.

Side C: Goes from the temple to initial point O, at 45-degree angle.

Because he moved directly east, making the temple directly north of current point O, sides A and B form a right angle.

Solving this way comes to 1414.

But this doesn't seem to jibe with an answer I found.

Let x = length of an edge of the square base = height of square-based pyramid. If we pass a plane through the apex of the pyramid perpendicular to the base, then the cross-section would appear as an isosceles triangle, with altitude from the vertex angle equal to x, the height of the pyramid. Let us consider half of the isosceles triangle, which is a right triangle whose legs are x and x/2 and the hypotenuse is 10 cm. Applying the Pythagorean Theorem, we have 10^2 = x^2 + (x/2)^2 100 = x^2 + x^2/4 100(4) = 4x^2 + x^2 400 = 5x^2 5x^2 = 400 x^2 = 400/5 x^2 = 80 x = sqrt(80) have you got what you want

imagine the starting point is O.

The temple is located at T.

and the second stop after walking 1000 meter to the East is S.

then you will have a triangle OST.

OS = 1000 meter.

while angle SOT is 45 degrees , it means ST is also 1000 meter.

then, the distance of the temple to O should be counted by using Phytagoras Theorema.

it should be square root of 2 x 1000^2

= square root of 2,000,000

=1,414.2135 metres.

good luck..

square root of 2 times 1000.

= 1414.21 Meters.

1000^2=2x^2

x=1000/rt2

=500rt2 m